| L(s) = 1 | − 1.67·2-s + 0.806·4-s − 5-s − 3.63·7-s + 1.99·8-s + 1.67·10-s + 1.61·11-s + 6.09·14-s − 4.96·16-s + 3.28·17-s + 7.11·19-s − 0.806·20-s − 2.70·22-s − 5.73·23-s + 25-s − 2.93·28-s − 7.44·29-s + 6.76·31-s + 4.31·32-s − 5.50·34-s + 3.63·35-s + 11.9·37-s − 11.9·38-s − 1.99·40-s − 5.13·41-s − 7.09·43-s + 1.29·44-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.403·4-s − 0.447·5-s − 1.37·7-s + 0.707·8-s + 0.529·10-s + 0.486·11-s + 1.62·14-s − 1.24·16-s + 0.797·17-s + 1.63·19-s − 0.180·20-s − 0.575·22-s − 1.19·23-s + 0.200·25-s − 0.554·28-s − 1.38·29-s + 1.21·31-s + 0.762·32-s − 0.944·34-s + 0.614·35-s + 1.96·37-s − 1.93·38-s − 0.316·40-s − 0.801·41-s − 1.08·43-s + 0.195·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6234568825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6234568825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 - 7.11T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 7.44T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 5.13T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 + 3.83T + 71T^{2} \) |
| 73 | \( 1 - 6.67T + 73T^{2} \) |
| 79 | \( 1 - 2.45T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 + 9.01T + 97T^{2} \) |
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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
−7.974181175346803841115197217665, −7.34515792768662245541163080743, −6.73832798906663163348869516153, −5.97933787761930808673107961037, −5.16296063832276056377896404576, −4.07962119752111403935939215914, −3.52204651462201114227550523443, −2.63740526914391991400091486243, −1.39074961581278494201531631678, −0.51518881308509546963082117726,
0.51518881308509546963082117726, 1.39074961581278494201531631678, 2.63740526914391991400091486243, 3.52204651462201114227550523443, 4.07962119752111403935939215914, 5.16296063832276056377896404576, 5.97933787761930808673107961037, 6.73832798906663163348869516153, 7.34515792768662245541163080743, 7.974181175346803841115197217665
