| L(s) = 1 | − 2.35·3-s + 1.88·5-s − 3.68·7-s + 2.55·9-s − 0.273·11-s + 6.67·13-s − 4.44·15-s + 17-s + 2.11·19-s + 8.68·21-s + 7.76·23-s − 1.44·25-s + 1.05·27-s + 5.02·29-s − 1.71·31-s + 0.643·33-s − 6.95·35-s − 0.885·37-s − 15.7·39-s + 9.14·41-s + 10.0·43-s + 4.81·45-s − 12.3·47-s + 6.58·49-s − 2.35·51-s − 9.87·53-s − 0.515·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.843·5-s − 1.39·7-s + 0.850·9-s − 0.0823·11-s + 1.85·13-s − 1.14·15-s + 0.242·17-s + 0.485·19-s + 1.89·21-s + 1.62·23-s − 0.288·25-s + 0.202·27-s + 0.932·29-s − 0.308·31-s + 0.112·33-s − 1.17·35-s − 0.145·37-s − 2.51·39-s + 1.42·41-s + 1.53·43-s + 0.717·45-s − 1.79·47-s + 0.940·49-s − 0.329·51-s − 1.35·53-s − 0.0694·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.346885721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346885721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 11 | \( 1 + 0.273T + 11T^{2} \) |
| 13 | \( 1 - 6.67T + 13T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 - 7.76T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 + 0.885T + 37T^{2} \) |
| 41 | \( 1 - 9.14T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 9.87T + 53T^{2} \) |
| 61 | \( 1 - 4.00T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 - 6.36T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.63578720254934880684537522070, −6.74587270945435113680675057777, −6.29364208036062538866468995129, −5.90558990310903795308392967225, −5.33073893144396467210381968237, −4.43496465498819998221596806357, −3.43293561959116497862607904493, −2.82754428765424430868733427812, −1.41208628919134178415768789892, −0.67736583913264722591388883945,
0.67736583913264722591388883945, 1.41208628919134178415768789892, 2.82754428765424430868733427812, 3.43293561959116497862607904493, 4.43496465498819998221596806357, 5.33073893144396467210381968237, 5.90558990310903795308392967225, 6.29364208036062538866468995129, 6.74587270945435113680675057777, 7.63578720254934880684537522070
