| L(s) = 1 | + 3.04·3-s − 3.52·5-s + 4.32·7-s + 6.26·9-s − 0.664·11-s + 1.58·13-s − 10.7·15-s − 7.29·17-s + 3.05·19-s + 13.1·21-s + 4.67·23-s + 7.40·25-s + 9.95·27-s + 4.75·29-s + 1.10·31-s − 2.02·33-s − 15.2·35-s + 10.9·37-s + 4.83·39-s − 8.90·41-s − 0.765·43-s − 22.0·45-s + 0.788·47-s + 11.6·49-s − 22.2·51-s − 5.96·53-s + 2.33·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 1.57·5-s + 1.63·7-s + 2.08·9-s − 0.200·11-s + 0.440·13-s − 2.76·15-s − 1.76·17-s + 0.701·19-s + 2.87·21-s + 0.975·23-s + 1.48·25-s + 1.91·27-s + 0.882·29-s + 0.199·31-s − 0.351·33-s − 2.57·35-s + 1.79·37-s + 0.773·39-s − 1.39·41-s − 0.116·43-s − 3.29·45-s + 0.115·47-s + 1.66·49-s − 3.10·51-s − 0.818·53-s + 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.532702728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.532702728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 + 0.664T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 - 4.67T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 + 0.765T + 43T^{2} \) |
| 47 | \( 1 - 0.788T + 47T^{2} \) |
| 53 | \( 1 + 5.96T + 53T^{2} \) |
| 59 | \( 1 - 0.649T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 - 5.78T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−8.394572812329865422905341491035, −7.943819147681844609730180694323, −7.35349815531813766007224080456, −6.66415308091941488854612781919, −4.94941103222792201336307318001, −4.49685158510243390054035180586, −3.83279766899530478368747794708, −2.98650049870251189098670181380, −2.15501759931403636393319747564, −1.05757071986759092587296530951,
1.05757071986759092587296530951, 2.15501759931403636393319747564, 2.98650049870251189098670181380, 3.83279766899530478368747794708, 4.49685158510243390054035180586, 4.94941103222792201336307318001, 6.66415308091941488854612781919, 7.35349815531813766007224080456, 7.943819147681844609730180694323, 8.394572812329865422905341491035
