| L(s) = 1 | − 1.75·2-s + 1.09·4-s + 1.58·5-s + 4.03·7-s + 1.59·8-s − 2.78·10-s + 3.16·11-s − 6.50·13-s − 7.09·14-s − 4.99·16-s − 2.47·17-s + 19-s + 1.73·20-s − 5.56·22-s − 2.42·23-s − 2.50·25-s + 11.4·26-s + 4.42·28-s + 7.50·29-s − 0.413·31-s + 5.59·32-s + 4.36·34-s + 6.37·35-s − 3.76·37-s − 1.75·38-s + 2.51·40-s + 2.25·41-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.547·4-s + 0.706·5-s + 1.52·7-s + 0.562·8-s − 0.879·10-s + 0.953·11-s − 1.80·13-s − 1.89·14-s − 1.24·16-s − 0.601·17-s + 0.229·19-s + 0.387·20-s − 1.18·22-s − 0.505·23-s − 0.500·25-s + 2.24·26-s + 0.835·28-s + 1.39·29-s − 0.0743·31-s + 0.989·32-s + 0.748·34-s + 1.07·35-s − 0.618·37-s − 0.285·38-s + 0.397·40-s + 0.351·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.75T + 2T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 7 | \( 1 - 4.03T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 + 6.50T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 + 0.413T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 2.25T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 0.751T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.54T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 0.608T + 97T^{2} \) |
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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
−7.81723776883196934077157418881, −6.94493481615504967407274709583, −6.41161877315167675797570825532, −5.18800668399118155948532310732, −4.82708758851110196467869792370, −4.09802011926557232366435856674, −2.63300591016124812463438332921, −1.84134380535214454968819487907, −1.36931859735605953040780217443, 0,
1.36931859735605953040780217443, 1.84134380535214454968819487907, 2.63300591016124812463438332921, 4.09802011926557232366435856674, 4.82708758851110196467869792370, 5.18800668399118155948532310732, 6.41161877315167675797570825532, 6.94493481615504967407274709583, 7.81723776883196934077157418881
