| L(s) = 1 | − 3·3-s − 19.8·5-s + 9·9-s − 23.9·11-s − 87.3·13-s + 59.6·15-s + 5.63·17-s + 64.8·19-s + 25.5·23-s + 270.·25-s − 27·27-s + 60.3·29-s − 122.·31-s + 71.8·33-s − 56.1·37-s + 262.·39-s + 299.·41-s + 501.·43-s − 179.·45-s + 305.·47-s − 16.9·51-s − 375.·53-s + 476.·55-s − 194.·57-s − 627.·59-s − 3.75·61-s + 1.73e3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.77·5-s + 0.333·9-s − 0.656·11-s − 1.86·13-s + 1.02·15-s + 0.0804·17-s + 0.783·19-s + 0.232·23-s + 2.16·25-s − 0.192·27-s + 0.386·29-s − 0.710·31-s + 0.378·33-s − 0.249·37-s + 1.07·39-s + 1.14·41-s + 1.77·43-s − 0.593·45-s + 0.948·47-s − 0.0464·51-s − 0.972·53-s + 1.16·55-s − 0.452·57-s − 1.38·59-s − 0.00788·61-s + 3.31·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 19.8T + 125T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.63T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.75T + 2.26e5T^{2} \) |
| 67 | \( 1 - 813.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 621.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 603.T + 9.12e5T^{2} \) |
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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
−7.898997975686862013606797065162, −7.49687571464462253156407366739, −7.05283319900711838563717376269, −5.77658190782031829464222694122, −4.88246375787487243494738098897, −4.40423799531124633245909996196, −3.36403122522750713576301559803, −2.47201365552535200657278568105, −0.790406109945933608839055154648, 0,
0.790406109945933608839055154648, 2.47201365552535200657278568105, 3.36403122522750713576301559803, 4.40423799531124633245909996196, 4.88246375787487243494738098897, 5.77658190782031829464222694122, 7.05283319900711838563717376269, 7.49687571464462253156407366739, 7.898997975686862013606797065162
