| L(s) = 1 | + 2.34·2-s + 3·3-s − 2.49·4-s + 15.3·5-s + 7.04·6-s − 10.1·7-s − 24.6·8-s + 9·9-s + 36.1·10-s + 15.0·11-s − 7.47·12-s − 23.7·14-s + 46.1·15-s − 37.8·16-s + 90.8·17-s + 21.1·18-s + 114.·19-s − 38.3·20-s − 30.3·21-s + 35.3·22-s + 75.7·23-s − 73.8·24-s + 112.·25-s + 27·27-s + 25.2·28-s + 214.·29-s + 108.·30-s + ⋯ |
| L(s) = 1 | + 0.829·2-s + 0.577·3-s − 0.311·4-s + 1.37·5-s + 0.479·6-s − 0.547·7-s − 1.08·8-s + 0.333·9-s + 1.14·10-s + 0.412·11-s − 0.179·12-s − 0.453·14-s + 0.795·15-s − 0.591·16-s + 1.29·17-s + 0.276·18-s + 1.38·19-s − 0.428·20-s − 0.315·21-s + 0.342·22-s + 0.686·23-s − 0.628·24-s + 0.897·25-s + 0.192·27-s + 0.170·28-s + 1.37·29-s + 0.659·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.093349902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.093349902\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.34T + 8T^{2} \) |
| 5 | \( 1 - 15.3T + 125T^{2} \) |
| 7 | \( 1 + 10.1T + 343T^{2} \) |
| 11 | \( 1 - 15.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 90.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 284.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 296.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 254.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 935.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 947.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 430.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 496.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 979.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 553.T + 9.12e5T^{2} \) |
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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
−10.16464420906629904367906438356, −9.521990497313230703376604054546, −9.084088113254200514204270549488, −7.76427467718713956689548572809, −6.50738806131106831740567495911, −5.71676827148595800550979347489, −4.89972544104242901965363877077, −3.52108869612884752454520474270, −2.77832230489755157097611157702, −1.19894248620003621435457753446,
1.19894248620003621435457753446, 2.77832230489755157097611157702, 3.52108869612884752454520474270, 4.89972544104242901965363877077, 5.71676827148595800550979347489, 6.50738806131106831740567495911, 7.76427467718713956689548572809, 9.084088113254200514204270549488, 9.521990497313230703376604054546, 10.16464420906629904367906438356
