| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 16-s − 17-s + 18-s − 20-s − 29-s + 32-s − 34-s + 36-s − 37-s − 40-s − 41-s − 45-s + 49-s − 53-s − 58-s − 61-s + 64-s − 68-s + 72-s − 73-s − 74-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 16-s − 17-s + 18-s − 20-s − 29-s + 32-s − 34-s + 36-s − 37-s − 40-s − 41-s − 45-s + 49-s − 53-s − 58-s − 61-s + 64-s − 68-s + 72-s − 73-s − 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.534361750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534361750\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−10.92120881172547309053773729877, −10.14019065025137438663238301407, −8.900316452865297839473977169278, −7.73535787900481635402714295316, −7.17008727396209882398305713201, −6.26525919184899939561990363410, −4.99159524111755786001663311271, −4.19759451080774027690421963894, −3.41657766923129214939689313907, −1.89531887771055503539822509181,
1.89531887771055503539822509181, 3.41657766923129214939689313907, 4.19759451080774027690421963894, 4.99159524111755786001663311271, 6.26525919184899939561990363410, 7.17008727396209882398305713201, 7.73535787900481635402714295316, 8.900316452865297839473977169278, 10.14019065025137438663238301407, 10.92120881172547309053773729877
