| L(s) = 1 | − 2.24·2-s + 3.04·4-s − 1.57·5-s − 2.20·7-s − 2.35·8-s + 3.53·10-s + 3.24·11-s − 4.93·13-s + 4.96·14-s − 0.801·16-s + 1.14·17-s − 19-s − 4.79·20-s − 7.29·22-s + 0.394·23-s − 2.52·25-s + 11.0·26-s − 6.73·28-s − 2.65·29-s + 2.64·31-s + 6.51·32-s − 2.57·34-s + 3.47·35-s + 1.38·37-s + 2.24·38-s + 3.70·40-s + 8.17·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.52·4-s − 0.703·5-s − 0.835·7-s − 0.833·8-s + 1.11·10-s + 0.979·11-s − 1.36·13-s + 1.32·14-s − 0.200·16-s + 0.277·17-s − 0.229·19-s − 1.07·20-s − 1.55·22-s + 0.0822·23-s − 0.504·25-s + 2.17·26-s − 1.27·28-s − 0.492·29-s + 0.475·31-s + 1.15·32-s − 0.441·34-s + 0.587·35-s + 0.227·37-s + 0.364·38-s + 0.586·40-s + 1.27·41-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 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 23 | \( 1 - 0.394T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 0.970T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 3.89T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 + 6.86T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
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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
−7.61909230387458384386818298350, −7.02246993171094558677875069459, −6.52593456091039944989972451482, −5.62659370150777796032527624810, −4.50632809344538285544646527902, −3.82784996315588437482520399851, −2.83876264915164775484686019842, −2.02720848257640575671940296118, −0.879201654812118362696909576014, 0,
0.879201654812118362696909576014, 2.02720848257640575671940296118, 2.83876264915164775484686019842, 3.82784996315588437482520399851, 4.50632809344538285544646527902, 5.62659370150777796032527624810, 6.52593456091039944989972451482, 7.02246993171094558677875069459, 7.61909230387458384386818298350
