| L(s) = 1 | + 5.51·2-s − 3.86·3-s + 22.4·4-s − 21.3·6-s + 79.7·8-s − 12.0·9-s + 34.5·11-s − 86.6·12-s − 68.8·13-s + 260.·16-s + 91.4·17-s − 66.7·18-s + 11.8·19-s + 190.·22-s + 0.104·23-s − 307.·24-s − 380.·26-s + 150.·27-s + 190.·29-s + 159.·31-s + 798.·32-s − 133.·33-s + 504.·34-s − 271.·36-s + 177.·37-s + 65.2·38-s + 265.·39-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 0.742·3-s + 2.80·4-s − 1.44·6-s + 3.52·8-s − 0.448·9-s + 0.946·11-s − 2.08·12-s − 1.46·13-s + 4.06·16-s + 1.30·17-s − 0.874·18-s + 0.142·19-s + 1.84·22-s + 0.000944·23-s − 2.61·24-s − 2.86·26-s + 1.07·27-s + 1.22·29-s + 0.925·31-s + 4.41·32-s − 0.702·33-s + 2.54·34-s − 1.25·36-s + 0.790·37-s + 0.278·38-s + 1.09·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.687631871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.687631871\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.51T + 8T^{2} \) |
| 3 | \( 1 + 3.86T + 27T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 91.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.104T + 1.21e4T^{2} \) |
| 29 | \( 1 - 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 8.25T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 353.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 240.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 778.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 151.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 311.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 639.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 391.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 473.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 839.T + 9.12e5T^{2} \) |
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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
−9.756060172754413278135152029089, −8.155534870711995651081002380062, −7.23475790673132912488234605967, −6.47710232626330434330160417739, −5.82770652251113556743941982517, −5.02981811494885418966576121344, −4.43635663819386642082709148160, −3.28383813907457420087770639570, −2.50331825335118487129376897344, −1.07681179736508049259330664310,
1.07681179736508049259330664310, 2.50331825335118487129376897344, 3.28383813907457420087770639570, 4.43635663819386642082709148160, 5.02981811494885418966576121344, 5.82770652251113556743941982517, 6.47710232626330434330160417739, 7.23475790673132912488234605967, 8.155534870711995651081002380062, 9.756060172754413278135152029089
