| L(s) = 1 | + 7.63·5-s − 5.63·7-s + 34.5·11-s + 13·13-s − 2·17-s + 88.1·19-s − 64·23-s − 66.7·25-s − 23.7·29-s + 284.·31-s − 42.9·35-s + 115.·37-s − 1.41·41-s + 337.·43-s − 198.·47-s − 311.·49-s − 59.0·53-s + 263.·55-s − 188.·59-s + 336.·61-s + 99.1·65-s + 531.·67-s − 510.·71-s − 164.·73-s − 194.·77-s + 29.3·79-s − 117.·83-s + ⋯ |
| L(s) = 1 | + 0.682·5-s − 0.303·7-s + 0.946·11-s + 0.277·13-s − 0.0285·17-s + 1.06·19-s − 0.580·23-s − 0.534·25-s − 0.152·29-s + 1.64·31-s − 0.207·35-s + 0.512·37-s − 0.00537·41-s + 1.19·43-s − 0.615·47-s − 0.907·49-s − 0.153·53-s + 0.645·55-s − 0.415·59-s + 0.707·61-s + 0.189·65-s + 0.968·67-s − 0.852·71-s − 0.263·73-s − 0.287·77-s + 0.0417·79-s − 0.155·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.775399349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.775399349\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 7.63T + 125T^{2} \) |
| 7 | \( 1 + 5.63T + 343T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64T + 1.21e4T^{2} \) |
| 29 | \( 1 + 23.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 284.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 1.41T + 6.89e4T^{2} \) |
| 43 | \( 1 - 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 59.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 188.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 336.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 29.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.989396734980124287727476614368, −8.127395019059551059899789966059, −7.25327983329754764604097199546, −6.30083719129566073539994324028, −5.89977264098585630320527784739, −4.79809035288805765310844686513, −3.86449315363557137551774524292, −2.91500596122087408259929638650, −1.80052841167742736213425197912, −0.807208050204740756171871700784,
0.807208050204740756171871700784, 1.80052841167742736213425197912, 2.91500596122087408259929638650, 3.86449315363557137551774524292, 4.79809035288805765310844686513, 5.89977264098585630320527784739, 6.30083719129566073539994324028, 7.25327983329754764604097199546, 8.127395019059551059899789966059, 8.989396734980124287727476614368
