| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 2·11-s + 8·13-s + 5·16-s + 10·17-s + 4·19-s − 6·20-s + 4·22-s + 2·23-s − 5·25-s + 16·26-s − 4·31-s + 6·32-s + 20·34-s + 8·37-s + 8·38-s − 8·40-s + 2·41-s + 6·44-s + 4·46-s + 10·47-s − 10·50-s + 24·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 0.603·11-s + 2.21·13-s + 5/4·16-s + 2.42·17-s + 0.917·19-s − 1.34·20-s + 0.852·22-s + 0.417·23-s − 25-s + 3.13·26-s − 0.718·31-s + 1.06·32-s + 3.42·34-s + 1.31·37-s + 1.29·38-s − 1.26·40-s + 0.312·41-s + 0.904·44-s + 0.589·46-s + 1.45·47-s − 1.41·50-s + 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.40199994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.40199994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 101 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 129 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 207 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 355 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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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
−7.74508030871712014958543519534, −7.65625881799531336998711208104, −7.09684585786297436029187355434, −6.77625810119612415077633585751, −6.21059010224548089020215803050, −6.13311897766382857179537988227, −5.72383782792806327150762259784, −5.62978403943114216456148467853, −4.95234323922384142734688426235, −4.86021416685617960625481715593, −4.19459137599238394008301547240, −3.91165538343535828654497052473, −3.54284797150618783803542455947, −3.53573991178784864703653255170, −3.09665745151793345288112964351, −2.67916067054928753832559166848, −1.87934919272965384754664742806, −1.59483563620952049126187521817, −0.880658531339775719151374883675, −0.847669397327943641882464577860,
0.847669397327943641882464577860, 0.880658531339775719151374883675, 1.59483563620952049126187521817, 1.87934919272965384754664742806, 2.67916067054928753832559166848, 3.09665745151793345288112964351, 3.53573991178784864703653255170, 3.54284797150618783803542455947, 3.91165538343535828654497052473, 4.19459137599238394008301547240, 4.86021416685617960625481715593, 4.95234323922384142734688426235, 5.62978403943114216456148467853, 5.72383782792806327150762259784, 6.13311897766382857179537988227, 6.21059010224548089020215803050, 6.77625810119612415077633585751, 7.09684585786297436029187355434, 7.65625881799531336998711208104, 7.74508030871712014958543519534
