| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 19-s + 23-s + 2·31-s − 32-s + 36-s − 37-s + 38-s − 41-s + 43-s − 46-s + 49-s + 53-s − 59-s − 2·62-s + 64-s − 72-s + 73-s + 74-s − 76-s − 79-s + 81-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 19-s + 23-s + 2·31-s − 32-s + 36-s − 37-s + 38-s − 41-s + 43-s − 46-s + 49-s + 53-s − 59-s − 2·62-s + 64-s − 72-s + 73-s + 74-s − 76-s − 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9066811631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9066811631\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.701303089833556414354202015606, −8.092941613099615660248479233757, −7.20600556366014599273509727807, −6.76411727303258126614293461928, −5.99201633122440725662820921769, −4.93097952423048382594548428502, −4.04240361668006629939702221022, −2.96719547574345053637329776374, −2.02792074836392834691044523196, −0.989895526482014255073297987569,
0.989895526482014255073297987569, 2.02792074836392834691044523196, 2.96719547574345053637329776374, 4.04240361668006629939702221022, 4.93097952423048382594548428502, 5.99201633122440725662820921769, 6.76411727303258126614293461928, 7.20600556366014599273509727807, 8.092941613099615660248479233757, 8.701303089833556414354202015606
