| L(s) = 1 | + 2-s + 4-s + 3.23·5-s + 7-s + 8-s + 3.23·10-s + 2·11-s + 0.763·13-s + 14-s + 16-s + 6.47·17-s − 19-s + 3.23·20-s + 2·22-s − 7.70·23-s + 5.47·25-s + 0.763·26-s + 28-s + 0.472·29-s − 5.70·31-s + 32-s + 6.47·34-s + 3.23·35-s + 5.23·37-s − 38-s + 3.23·40-s − 10.9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.44·5-s + 0.377·7-s + 0.353·8-s + 1.02·10-s + 0.603·11-s + 0.211·13-s + 0.267·14-s + 0.250·16-s + 1.56·17-s − 0.229·19-s + 0.723·20-s + 0.426·22-s − 1.60·23-s + 1.09·25-s + 0.149·26-s + 0.188·28-s + 0.0876·29-s − 1.02·31-s + 0.176·32-s + 1.10·34-s + 0.546·35-s + 0.860·37-s − 0.162·38-s + 0.511·40-s − 1.70·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.150431138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.150431138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 1.52T + 97T^{2} \) |
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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
−9.073349489214038813066459231311, −8.123831019162914718383092756176, −7.32296905661295268138451525865, −6.27189929775602511037900967687, −5.86620657714985254945901707436, −5.18153571584018487461449497642, −4.16215212490049906011557815255, −3.24003594589147666818476208969, −2.09860271302388390178485610003, −1.38788186723607975211791770465,
1.38788186723607975211791770465, 2.09860271302388390178485610003, 3.24003594589147666818476208969, 4.16215212490049906011557815255, 5.18153571584018487461449497642, 5.86620657714985254945901707436, 6.27189929775602511037900967687, 7.32296905661295268138451525865, 8.123831019162914718383092756176, 9.073349489214038813066459231311
