| L(s) = 1 | + 1.21·3-s − 20.5·5-s − 7·7-s − 25.5·9-s + 11·11-s + 66.7·13-s − 24.9·15-s − 59.7·17-s + 131.·19-s − 8.49·21-s + 163.·23-s + 296.·25-s − 63.7·27-s − 77.2·29-s + 206.·31-s + 13.3·33-s + 143.·35-s − 215.·37-s + 81.0·39-s − 403.·41-s + 43.5·43-s + 523.·45-s + 69.1·47-s + 49·49-s − 72.5·51-s + 334.·53-s − 225.·55-s + ⋯ |
| L(s) = 1 | + 0.233·3-s − 1.83·5-s − 0.377·7-s − 0.945·9-s + 0.301·11-s + 1.42·13-s − 0.429·15-s − 0.852·17-s + 1.58·19-s − 0.0883·21-s + 1.47·23-s + 2.37·25-s − 0.454·27-s − 0.494·29-s + 1.19·31-s + 0.0704·33-s + 0.693·35-s − 0.956·37-s + 0.332·39-s − 1.53·41-s + 0.154·43-s + 1.73·45-s + 0.214·47-s + 0.142·49-s − 0.199·51-s + 0.868·53-s − 0.553·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.192708198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192708198\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 1.21T + 27T^{2} \) |
| 5 | \( 1 + 20.5T + 125T^{2} \) |
| 13 | \( 1 - 66.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 77.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 403.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 43.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 69.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 334.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 681.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 24.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 285.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 522.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 592.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 92.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 308.T + 9.12e5T^{2} \) |
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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
−11.60161726400676964833911311169, −10.58990186514565611189235454138, −8.962639477443248122775964784736, −8.542620130845934037743185315858, −7.48599983116517044300056620306, −6.55958084575416216877764416892, −5.09875930191017262695133465363, −3.74389064647217801046790785382, −3.12991117196745640994832449933, −0.74670210453715653109734864113,
0.74670210453715653109734864113, 3.12991117196745640994832449933, 3.74389064647217801046790785382, 5.09875930191017262695133465363, 6.55958084575416216877764416892, 7.48599983116517044300056620306, 8.542620130845934037743185315858, 8.962639477443248122775964784736, 10.58990186514565611189235454138, 11.60161726400676964833911311169
