| L(s) = 1 | + 2-s + 4-s − 1.58·5-s + 8-s − 1.58·10-s + 11-s − 4.24·13-s + 16-s − 2.82·17-s − 0.171·19-s − 1.58·20-s + 22-s + 3.24·23-s − 2.48·25-s − 4.24·26-s − 8.24·29-s + 1.24·31-s + 32-s − 2.82·34-s − 3.24·37-s − 0.171·38-s − 1.58·40-s − 11.8·41-s + 10.4·43-s + 44-s + 3.24·46-s − 0.343·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.709·5-s + 0.353·8-s − 0.501·10-s + 0.301·11-s − 1.17·13-s + 0.250·16-s − 0.685·17-s − 0.0393·19-s − 0.354·20-s + 0.213·22-s + 0.676·23-s − 0.497·25-s − 0.832·26-s − 1.53·29-s + 0.223·31-s + 0.176·32-s − 0.485·34-s − 0.533·37-s − 0.0278·38-s − 0.250·40-s − 1.84·41-s + 1.59·43-s + 0.150·44-s + 0.478·46-s − 0.0500·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 0.171T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 0.343T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.399437985115563856040938523842, −7.41748661564723328972789149222, −7.11263484800049444606744803088, −6.10236072449533074926939632011, −5.21641128710125560001361039286, −4.47744199359416047185517411990, −3.74477253681978377587582294967, −2.81351938430334120966769335143, −1.76189166197775935190942078664, 0,
1.76189166197775935190942078664, 2.81351938430334120966769335143, 3.74477253681978377587582294967, 4.47744199359416047185517411990, 5.21641128710125560001361039286, 6.10236072449533074926939632011, 7.11263484800049444606744803088, 7.41748661564723328972789149222, 8.399437985115563856040938523842
