| L(s) = 1 | + 2.08·2-s + 2.35·4-s − 5-s + 4.08·7-s + 0.734·8-s − 2.08·10-s + 1.35·11-s + 0.648·13-s + 8.52·14-s − 3.17·16-s + 1.35·17-s + 0.648·19-s − 2.35·20-s + 2.82·22-s − 4.79·23-s + 25-s + 1.35·26-s + 9.61·28-s − 3.87·29-s − 7.69·31-s − 8.08·32-s + 2.82·34-s − 4.08·35-s + 7.52·37-s + 1.35·38-s − 0.734·40-s + 0.179·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s − 0.447·5-s + 1.54·7-s + 0.259·8-s − 0.659·10-s + 0.407·11-s + 0.179·13-s + 2.27·14-s − 0.793·16-s + 0.327·17-s + 0.148·19-s − 0.525·20-s + 0.601·22-s − 0.998·23-s + 0.200·25-s + 0.265·26-s + 1.81·28-s − 0.719·29-s − 1.38·31-s − 1.42·32-s + 0.483·34-s − 0.690·35-s + 1.23·37-s + 0.219·38-s − 0.116·40-s + 0.0280·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.985129585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.985129585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 - 0.648T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 0.179T + 41T^{2} \) |
| 43 | \( 1 + 0.820T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 + 4.17T + 59T^{2} \) |
| 61 | \( 1 + 3.82T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−11.45359683933688728953493631087, −10.94163411239748998932598819198, −9.421856993621271807439932325262, −8.244814752793429897264647805926, −7.44243333622413619716615056519, −6.17699643520193400866159855186, −5.21494526850361755671440842486, −4.39699754656650699664459128108, −3.49752636174282218812187065946, −1.89342420448833102084428967755,
1.89342420448833102084428967755, 3.49752636174282218812187065946, 4.39699754656650699664459128108, 5.21494526850361755671440842486, 6.17699643520193400866159855186, 7.44243333622413619716615056519, 8.244814752793429897264647805926, 9.421856993621271807439932325262, 10.94163411239748998932598819198, 11.45359683933688728953493631087
