| L(s) = 1 | − 2·3-s − 2·4-s + 4·5-s − 4·7-s + 2·8-s − 2·9-s + 4·12-s + 16·13-s − 8·15-s + 3·16-s + 2·17-s + 6·19-s − 8·20-s + 8·21-s − 2·23-s − 4·24-s + 10·25-s + 12·27-s + 8·28-s + 12·29-s − 6·31-s − 4·32-s − 16·35-s + 4·36-s − 6·37-s − 32·39-s + 8·40-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 1.78·5-s − 1.51·7-s + 0.707·8-s − 2/3·9-s + 1.15·12-s + 4.43·13-s − 2.06·15-s + 3/4·16-s + 0.485·17-s + 1.37·19-s − 1.78·20-s + 1.74·21-s − 0.417·23-s − 0.816·24-s + 2·25-s + 2.30·27-s + 1.51·28-s + 2.22·29-s − 1.07·31-s − 0.707·32-s − 2.70·35-s + 2/3·36-s − 0.986·37-s − 5.12·39-s + 1.26·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.928248516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.928248516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p T^{2} - p T^{3} + T^{4} - p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 2 T + 2 p T^{2} + 4 T^{3} + 4 p T^{4} + 4 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 16 T + 122 T^{2} - 610 T^{3} + 184 p T^{4} - 610 p T^{5} + 122 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 18 T^{2} - 4 T^{3} + 280 T^{4} - 4 p T^{5} + 18 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 6 T + 56 T^{2} - 214 T^{3} + 1270 T^{4} - 214 p T^{5} + 56 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 84 T^{2} + 130 T^{3} + 2818 T^{4} + 130 p T^{5} + 84 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 12 T + 100 T^{2} - 740 T^{3} + 4742 T^{4} - 740 p T^{5} + 100 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 6 T + 88 T^{2} + 392 T^{3} + 3432 T^{4} + 392 p T^{5} + 88 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 6 T + 4 T^{2} - 214 T^{3} - 806 T^{4} - 214 p T^{5} + 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 18 T + 224 T^{2} - 1892 T^{3} + 13708 T^{4} - 1892 p T^{5} + 224 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 6 T + 100 T^{2} - 558 T^{3} + 5874 T^{4} - 558 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 4 T + 58 T^{2} + 138 T^{3} + 772 T^{4} + 138 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 34 T + 628 T^{2} - 7566 T^{3} + 64986 T^{4} - 7566 p T^{5} + 628 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 10 T + 140 T^{2} + 1196 T^{3} + 8856 T^{4} + 1196 p T^{5} + 140 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 8 T + 148 T^{2} + 438 T^{3} + 8548 T^{4} + 438 p T^{5} + 148 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 14 T + 256 T^{2} - 2414 T^{3} + 25642 T^{4} - 2414 p T^{5} + 256 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 164 T^{2} + 160 T^{3} + 15078 T^{4} + 160 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 2 T + 170 T^{2} - 740 T^{3} + 14488 T^{4} - 740 p T^{5} + 170 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 8 T + 288 T^{2} + 1524 T^{3} + 32234 T^{4} + 1524 p T^{5} + 288 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 10 T + 76 T^{2} + 282 T^{3} - 2466 T^{4} + 282 p T^{5} + 76 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 10 T + 220 T^{2} - 1846 T^{3} + 28766 T^{4} - 1846 p T^{5} + 220 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 34 T + 768 T^{2} + 11542 T^{3} + 132342 T^{4} + 11542 p T^{5} + 768 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.81279724437249517959331384158, −5.77073159425482995155993174683, −5.66498368436371605570367031470, −5.52310260548290079744077402957, −5.47587606974518306400603609989, −4.85545690593474159161484483684, −4.84595392973163437482729397727, −4.72754198882735729813325736928, −4.23774074174434625104239095415, −4.07236619560354025355727151532, −3.87311113499332313420918435825, −3.77262291143616836897462179992, −3.67817805125701269122832100890, −3.13513877323574697811514836450, −3.07707964739743253237642831444, −2.94654779255398909215649967950, −2.57026330281682078671834502480, −2.46494241322341450304737426759, −2.10959477355501321692939124671, −1.49755394304438504762182975658, −1.42369834187236349080823586888, −1.14033807365131346211276974213, −1.00267425765558277668241360602, −0.58674563758804923125089937371, −0.57095863514621023904663267088,
0.57095863514621023904663267088, 0.58674563758804923125089937371, 1.00267425765558277668241360602, 1.14033807365131346211276974213, 1.42369834187236349080823586888, 1.49755394304438504762182975658, 2.10959477355501321692939124671, 2.46494241322341450304737426759, 2.57026330281682078671834502480, 2.94654779255398909215649967950, 3.07707964739743253237642831444, 3.13513877323574697811514836450, 3.67817805125701269122832100890, 3.77262291143616836897462179992, 3.87311113499332313420918435825, 4.07236619560354025355727151532, 4.23774074174434625104239095415, 4.72754198882735729813325736928, 4.84595392973163437482729397727, 4.85545690593474159161484483684, 5.47587606974518306400603609989, 5.52310260548290079744077402957, 5.66498368436371605570367031470, 5.77073159425482995155993174683, 5.81279724437249517959331384158
