| L(s) = 1 | − 2.34·3-s + 3.03·5-s + 3.94·7-s + 2.52·9-s − 11-s + 2.12·13-s − 7.12·15-s + 5.59·17-s + 19-s − 9.27·21-s + 2.55·23-s + 4.20·25-s + 1.12·27-s + 7.80·29-s − 10.6·31-s + 2.34·33-s + 11.9·35-s + 7.46·37-s − 4.98·39-s − 9.34·41-s + 8.94·43-s + 7.64·45-s + 2.06·47-s + 8.58·49-s − 13.1·51-s + 11.2·53-s − 3.03·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 1.35·5-s + 1.49·7-s + 0.840·9-s − 0.301·11-s + 0.588·13-s − 1.84·15-s + 1.35·17-s + 0.229·19-s − 2.02·21-s + 0.531·23-s + 0.841·25-s + 0.216·27-s + 1.44·29-s − 1.90·31-s + 0.409·33-s + 2.02·35-s + 1.22·37-s − 0.798·39-s − 1.45·41-s + 1.36·43-s + 1.14·45-s + 0.301·47-s + 1.22·49-s − 1.84·51-s + 1.54·53-s − 0.409·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040084859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040084859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 - 5.59T + 17T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 2.06T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 4.01T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 + 6.32T + 79T^{2} \) |
| 83 | \( 1 - 6.77T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.650391672439976780354154588542, −7.79876285979527099399519988223, −7.01742831456920961829241333618, −6.02906073484056087077581821535, −5.58156898615236197835182646171, −5.14876273916357808120374044916, −4.31145299372251655446178819059, −2.87598102298463329051879212618, −1.66471697471263570970087506107, −1.02703220699927617873917194693,
1.02703220699927617873917194693, 1.66471697471263570970087506107, 2.87598102298463329051879212618, 4.31145299372251655446178819059, 5.14876273916357808120374044916, 5.58156898615236197835182646171, 6.02906073484056087077581821535, 7.01742831456920961829241333618, 7.79876285979527099399519988223, 8.650391672439976780354154588542
