| L(s) = 1 | − 4.88·2-s + 15.8·4-s + 15.7·5-s + 24.1·7-s − 38.1·8-s − 76.8·10-s − 42.5·11-s − 74.5·13-s − 118.·14-s + 59.6·16-s − 18.5·17-s + 79.2·19-s + 249.·20-s + 207.·22-s − 195.·23-s + 123.·25-s + 363.·26-s + 382.·28-s − 125.·29-s + 109.·31-s + 14.2·32-s + 90.5·34-s + 381.·35-s + 397.·37-s − 386.·38-s − 600.·40-s + 7.28·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.97·4-s + 1.40·5-s + 1.30·7-s − 1.68·8-s − 2.43·10-s − 1.16·11-s − 1.58·13-s − 2.25·14-s + 0.931·16-s − 0.264·17-s + 0.956·19-s + 2.78·20-s + 2.01·22-s − 1.77·23-s + 0.985·25-s + 2.74·26-s + 2.58·28-s − 0.801·29-s + 0.635·31-s + 0.0785·32-s + 0.456·34-s + 1.84·35-s + 1.76·37-s − 1.65·38-s − 2.37·40-s + 0.0277·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 211 | \( 1 + 211T \) |
| good | 2 | \( 1 + 4.88T + 8T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 7 | \( 1 - 24.1T + 343T^{2} \) |
| 11 | \( 1 + 42.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 74.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 397.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 7.28T + 6.89e4T^{2} \) |
| 43 | \( 1 - 61.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 322.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 500.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 160.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 435.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 572.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 902.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 498.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 89.9T + 9.12e5T^{2} \) |
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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
−8.469990427745401279208421875172, −7.69861702210156332398405358351, −7.45138380859196138077644815771, −6.18255001451381801737913635850, −5.43918649294520245243136390907, −4.61661090261872840668156287356, −2.49850297511181506804980750796, −2.22403980154387721628080328701, −1.25875542131730929555948658666, 0,
1.25875542131730929555948658666, 2.22403980154387721628080328701, 2.49850297511181506804980750796, 4.61661090261872840668156287356, 5.43918649294520245243136390907, 6.18255001451381801737913635850, 7.45138380859196138077644815771, 7.69861702210156332398405358351, 8.469990427745401279208421875172
