| L(s) = 1 | + 4.08·2-s − 9.09·3-s + 8.65·4-s + 5·5-s − 37.0·6-s − 18.7·7-s + 2.65·8-s + 55.6·9-s + 20.4·10-s − 11·11-s − 78.6·12-s + 21.8·13-s − 76.4·14-s − 45.4·15-s − 58.3·16-s − 85.6·17-s + 227.·18-s + 19·19-s + 43.2·20-s + 170.·21-s − 44.8·22-s + 44.7·23-s − 24.1·24-s + 25·25-s + 89.2·26-s − 260.·27-s − 161.·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s − 1.74·3-s + 1.08·4-s + 0.447·5-s − 2.52·6-s − 1.01·7-s + 0.117·8-s + 2.06·9-s + 0.645·10-s − 0.301·11-s − 1.89·12-s + 0.466·13-s − 1.45·14-s − 0.782·15-s − 0.912·16-s − 1.22·17-s + 2.97·18-s + 0.229·19-s + 0.483·20-s + 1.76·21-s − 0.434·22-s + 0.405·23-s − 0.205·24-s + 0.200·25-s + 0.673·26-s − 1.85·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.746392781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746392781\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 4.08T + 8T^{2} \) |
| 3 | \( 1 + 9.09T + 27T^{2} \) |
| 7 | \( 1 + 18.7T + 343T^{2} \) |
| 13 | \( 1 - 21.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.6T + 4.91e3T^{2} \) |
| 23 | \( 1 - 44.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 279.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 564.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 583.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 287.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 427.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 854.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 19.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 629.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 161.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 184.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 323.T + 9.12e5T^{2} \) |
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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
−9.825368773878443971382961250022, −8.923856271909362895506120567338, −7.14631810176876342343328964015, −6.55795533047606581414383644176, −5.92493379097436100497440809747, −5.35055393824778479978799310612, −4.49846064389196737434691940079, −3.61354973329038940986944742872, −2.29057548403783393422757946065, −0.59144046477168163669491202687,
0.59144046477168163669491202687, 2.29057548403783393422757946065, 3.61354973329038940986944742872, 4.49846064389196737434691940079, 5.35055393824778479978799310612, 5.92493379097436100497440809747, 6.55795533047606581414383644176, 7.14631810176876342343328964015, 8.923856271909362895506120567338, 9.825368773878443971382961250022
