L(s) = 1 | + 2.48·2-s − 3-s + 4.16·4-s − 5-s − 2.48·6-s − 1.71·7-s + 5.37·8-s + 9-s − 2.48·10-s + 5.37·11-s − 4.16·12-s − 13-s − 4.26·14-s + 15-s + 5.01·16-s + 1.14·17-s + 2.48·18-s + 4.76·19-s − 4.16·20-s + 1.71·21-s + 13.3·22-s − 4.06·23-s − 5.37·24-s + 25-s − 2.48·26-s − 27-s − 7.15·28-s + ⋯ |
L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.08·4-s − 0.447·5-s − 1.01·6-s − 0.649·7-s + 1.90·8-s + 0.333·9-s − 0.785·10-s + 1.62·11-s − 1.20·12-s − 0.277·13-s − 1.13·14-s + 0.258·15-s + 1.25·16-s + 0.277·17-s + 0.585·18-s + 1.09·19-s − 0.931·20-s + 0.374·21-s + 2.84·22-s − 0.846·23-s − 1.09·24-s + 0.200·25-s − 0.486·26-s − 0.192·27-s − 1.35·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.812392919\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.812392919\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 37 | \( 1 + 9.64T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.50T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 0.173T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 2.41T + 73T^{2} \) |
| 79 | \( 1 - 5.90T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.51680372754981128566510902616, −7.11726566135711239869221135437, −6.41763702854850943210473493022, −5.80994434571279174155949859542, −5.27277718273329587064721978099, −4.21601019879465536124105742521, −3.92314891731835723856409202308, −3.18926946820696469383244905964, −2.15568319818460888849274506961, −0.931741627481619710012024255848,
0.931741627481619710012024255848, 2.15568319818460888849274506961, 3.18926946820696469383244905964, 3.92314891731835723856409202308, 4.21601019879465536124105742521, 5.27277718273329587064721978099, 5.80994434571279174155949859542, 6.41763702854850943210473493022, 7.11726566135711239869221135437, 7.51680372754981128566510902616