| L(s) = 1 | + 2·2-s + 3·3-s − 6·5-s + 6·6-s − 8·8-s − 12·10-s + 30·11-s − 106·13-s − 18·15-s − 16·16-s − 84·17-s − 97·19-s + 60·22-s − 84·23-s − 24·24-s + 125·25-s − 212·26-s − 27·27-s − 360·29-s − 36·30-s + 179·31-s + 90·33-s − 168·34-s + 145·37-s − 194·38-s − 318·39-s + 48·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.536·5-s + 0.408·6-s − 0.353·8-s − 0.379·10-s + 0.822·11-s − 2.26·13-s − 0.309·15-s − 1/4·16-s − 1.19·17-s − 1.17·19-s + 0.581·22-s − 0.761·23-s − 0.204·24-s + 25-s − 1.59·26-s − 0.192·27-s − 2.30·29-s − 0.219·30-s + 1.03·31-s + 0.474·33-s − 0.847·34-s + 0.644·37-s − 0.828·38-s − 1.30·39-s + 0.189·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.112792626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112792626\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 53 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 97 T + 2550 T^{2} + 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 179 T + 2250 T^{2} - 179 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 145 T - 29628 T^{2} - 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 325 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 366 T + 30133 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 768 T + 440947 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 264 T - 135683 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 818 T + 442143 T^{2} - 818 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 523 T - 27234 T^{2} - 523 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 43 T - 387168 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1171 T + 878202 T^{2} - 1171 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 810 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 600 T - 344969 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 386 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.64412085671395109896002696873, −11.32635511014385337495089751103, −10.71731277542905959397216823196, −9.948213639733258967646614336823, −9.809996676667652608263405024575, −9.173443539561046308474640757567, −8.695430511141415806803508417600, −8.255279729075972079095142687935, −7.75973394142389535973324429192, −7.02165032012570114678519278852, −6.77653210343503749617671502384, −6.24330162926711574691797188465, −5.27645817280584977826617447898, −4.93341377206260618906589441098, −4.30948530926215371258938081246, −3.86978267469246152517173385039, −3.20729547127759852639819060712, −2.31897163716872579123307128941, −1.97283841003663071407784653991, −0.31737253894562537102136508648,
0.31737253894562537102136508648, 1.97283841003663071407784653991, 2.31897163716872579123307128941, 3.20729547127759852639819060712, 3.86978267469246152517173385039, 4.30948530926215371258938081246, 4.93341377206260618906589441098, 5.27645817280584977826617447898, 6.24330162926711574691797188465, 6.77653210343503749617671502384, 7.02165032012570114678519278852, 7.75973394142389535973324429192, 8.255279729075972079095142687935, 8.695430511141415806803508417600, 9.173443539561046308474640757567, 9.809996676667652608263405024575, 9.948213639733258967646614336823, 10.71731277542905959397216823196, 11.32635511014385337495089751103, 11.64412085671395109896002696873
