| L(s) = 1 | − 3·3-s − 2·5-s − 7·7-s + 9·9-s + 8·11-s + 42·13-s + 6·15-s − 2·17-s + 124·19-s + 21·21-s + 76·23-s − 121·25-s − 27·27-s − 254·29-s − 72·31-s − 24·33-s + 14·35-s − 398·37-s − 126·39-s + 462·41-s − 212·43-s − 18·45-s − 264·47-s + 49·49-s + 6·51-s + 162·53-s − 16·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.178·5-s − 0.377·7-s + 1/3·9-s + 0.219·11-s + 0.896·13-s + 0.103·15-s − 0.0285·17-s + 1.49·19-s + 0.218·21-s + 0.689·23-s − 0.967·25-s − 0.192·27-s − 1.62·29-s − 0.417·31-s − 0.126·33-s + 0.0676·35-s − 1.76·37-s − 0.517·39-s + 1.75·41-s − 0.751·43-s − 0.0596·45-s − 0.819·47-s + 1/7·49-s + 0.0164·51-s + 0.419·53-s − 0.0392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.500631551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500631551\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 76 T + p^{3} T^{2} \) |
| 29 | \( 1 + 254 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 462 T + p^{3} T^{2} \) |
| 43 | \( 1 + 212 T + p^{3} T^{2} \) |
| 47 | \( 1 + 264 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 772 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 236 T + p^{3} T^{2} \) |
| 73 | \( 1 - 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1190 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.377566873177240010680911729369, −8.463483541318776965749636607769, −7.44483352704001028430906237036, −6.83987316622874127837787707642, −5.78672464866936143945244738395, −5.27163876377852061747349790731, −3.96495852713497639523618853738, −3.31696072332016564908850277595, −1.78694931127548525177286581025, −0.64380887177876923145162469506,
0.64380887177876923145162469506, 1.78694931127548525177286581025, 3.31696072332016564908850277595, 3.96495852713497639523618853738, 5.27163876377852061747349790731, 5.78672464866936143945244738395, 6.83987316622874127837787707642, 7.44483352704001028430906237036, 8.463483541318776965749636607769, 9.377566873177240010680911729369
