| L(s) = 1 | + 2.89·3-s + 2.58·5-s + 0.456·7-s + 5.39·9-s + 4.71·11-s + 5.34·13-s + 7.49·15-s − 2.87·17-s + 19-s + 1.32·21-s − 8.48·23-s + 1.68·25-s + 6.94·27-s − 7.59·29-s − 0.528·31-s + 13.6·33-s + 1.17·35-s + 10.4·37-s + 15.4·39-s − 7.79·41-s + 6.23·43-s + 13.9·45-s − 4.40·47-s − 6.79·49-s − 8.33·51-s + 53-s + 12.1·55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 1.15·5-s + 0.172·7-s + 1.79·9-s + 1.42·11-s + 1.48·13-s + 1.93·15-s − 0.697·17-s + 0.229·19-s + 0.288·21-s − 1.76·23-s + 0.336·25-s + 1.33·27-s − 1.41·29-s − 0.0949·31-s + 2.37·33-s + 0.199·35-s + 1.71·37-s + 2.48·39-s − 1.21·41-s + 0.951·43-s + 2.07·45-s − 0.643·47-s − 0.970·49-s − 1.16·51-s + 0.137·53-s + 1.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.151551841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.151551841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 - 0.456T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 + 0.528T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 - 0.942T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 + 0.382T + 71T^{2} \) |
| 73 | \( 1 + 1.20T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 8.86T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−8.491355123271103505527685312659, −8.003180830528679073658019639417, −7.01030596149171214548598679337, −6.24917059352526750305265800546, −5.70248970677474979845311238337, −4.18099320280607045704310054199, −3.89681521767092916101891156845, −2.88650924019359481094657407186, −1.85344198714232254709644486162, −1.51041386674584030760683134326,
1.51041386674584030760683134326, 1.85344198714232254709644486162, 2.88650924019359481094657407186, 3.89681521767092916101891156845, 4.18099320280607045704310054199, 5.70248970677474979845311238337, 6.24917059352526750305265800546, 7.01030596149171214548598679337, 8.003180830528679073658019639417, 8.491355123271103505527685312659
