| L(s) = 1 | + 3-s + 5·5-s + 7·7-s − 8·9-s + 13·11-s + 19·13-s + 5·15-s − 29·17-s + 7·21-s + 25·25-s − 17·27-s + 23·29-s + 13·33-s + 35·35-s + 19·39-s − 40·45-s − 31·47-s + 49·49-s − 29·51-s + 65·55-s − 56·63-s + 95·65-s − 2·71-s + 34·73-s + 25·75-s + 91·77-s + 157·79-s + ⋯ |
| L(s) = 1 | + 1/3·3-s + 5-s + 7-s − 8/9·9-s + 1.18·11-s + 1.46·13-s + 1/3·15-s − 1.70·17-s + 1/3·21-s + 25-s − 0.629·27-s + 0.793·29-s + 0.393·33-s + 35-s + 0.487·39-s − 8/9·45-s − 0.659·47-s + 49-s − 0.568·51-s + 1.18·55-s − 8/9·63-s + 1.46·65-s − 0.0281·71-s + 0.465·73-s + 1/3·75-s + 1.18·77-s + 1.98·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.722681966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722681966\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - T + p^{2} T^{2} \) |
| 11 | \( 1 - 13 T + p^{2} T^{2} \) |
| 13 | \( 1 - 19 T + p^{2} T^{2} \) |
| 17 | \( 1 + 29 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 23 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 31 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( 1 + 2 T + p^{2} T^{2} \) |
| 73 | \( 1 - 34 T + p^{2} T^{2} \) |
| 79 | \( 1 - 157 T + p^{2} T^{2} \) |
| 83 | \( 1 - 86 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 149 T + p^{2} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.81774139496893165209540094579, −9.419704847457727036852785059137, −8.780413284014397940452448969180, −8.247705902108198010203357563341, −6.68960528705947591122207616241, −6.10040190795776390739412804847, −4.96712730622036284748748766932, −3.82847740500390756064826557597, −2.39859128913470652638872784326, −1.33088168508529191997763161761,
1.33088168508529191997763161761, 2.39859128913470652638872784326, 3.82847740500390756064826557597, 4.96712730622036284748748766932, 6.10040190795776390739412804847, 6.68960528705947591122207616241, 8.247705902108198010203357563341, 8.780413284014397940452448969180, 9.419704847457727036852785059137, 10.81774139496893165209540094579
