| L(s) = 1 | − 1.58·2-s + 0.500·4-s − 1.67·5-s + 1.85·7-s + 2.37·8-s + 2.64·10-s − 5.15·11-s − 1.44·13-s − 2.93·14-s − 4.75·16-s + 5.00·17-s − 19-s − 0.835·20-s + 8.14·22-s + 7.10·23-s − 2.20·25-s + 2.27·26-s + 0.927·28-s + 3.99·29-s − 6.67·31-s + 2.76·32-s − 7.92·34-s − 3.10·35-s − 12.0·37-s + 1.58·38-s − 3.96·40-s + 2.21·41-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.250·4-s − 0.747·5-s + 0.701·7-s + 0.838·8-s + 0.835·10-s − 1.55·11-s − 0.399·13-s − 0.783·14-s − 1.18·16-s + 1.21·17-s − 0.229·19-s − 0.186·20-s + 1.73·22-s + 1.48·23-s − 0.441·25-s + 0.446·26-s + 0.175·28-s + 0.742·29-s − 1.19·31-s + 0.489·32-s − 1.35·34-s − 0.524·35-s − 1.98·37-s + 0.256·38-s − 0.626·40-s + 0.345·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5829469489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5829469489\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.58T + 2T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 23 | \( 1 - 7.10T + 23T^{2} \) |
| 29 | \( 1 - 3.99T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 + 0.115T + 43T^{2} \) |
| 53 | \( 1 - 8.83T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 - 4.55T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.85T + 79T^{2} \) |
| 83 | \( 1 + 7.26T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.948873217042742356532353469247, −7.33759521574617808231016792447, −7.01010285249916869705363200544, −5.43591962790931687701704532739, −5.20860344319138435227162322101, −4.34460145444346044504185769075, −3.43128040900421347712252620805, −2.48907397823613906451706781518, −1.50306135541257883432071575142, −0.46784168067959400810791712169,
0.46784168067959400810791712169, 1.50306135541257883432071575142, 2.48907397823613906451706781518, 3.43128040900421347712252620805, 4.34460145444346044504185769075, 5.20860344319138435227162322101, 5.43591962790931687701704532739, 7.01010285249916869705363200544, 7.33759521574617808231016792447, 7.948873217042742356532353469247
