| L(s) = 1 | + 2.07·2-s + 3-s + 2.31·4-s − 2.30·5-s + 2.07·6-s + 0.652·8-s + 9-s − 4.77·10-s − 2.24·11-s + 2.31·12-s + 4.05·13-s − 2.30·15-s − 3.27·16-s + 1.22·17-s + 2.07·18-s + 4.41·19-s − 5.32·20-s − 4.65·22-s + 6.56·23-s + 0.652·24-s + 0.291·25-s + 8.42·26-s + 27-s + 3.41·29-s − 4.77·30-s − 1.07·31-s − 8.10·32-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 0.577·3-s + 1.15·4-s − 1.02·5-s + 0.847·6-s + 0.230·8-s + 0.333·9-s − 1.51·10-s − 0.675·11-s + 0.668·12-s + 1.12·13-s − 0.593·15-s − 0.818·16-s + 0.297·17-s + 0.489·18-s + 1.01·19-s − 1.19·20-s − 0.992·22-s + 1.36·23-s + 0.133·24-s + 0.0583·25-s + 1.65·26-s + 0.192·27-s + 0.634·29-s − 0.872·30-s − 0.193·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.747216512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.747216512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 5.24T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 3.55T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 97 | \( 1 + 3.88T + 97T^{2} \) |
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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
−7.967064882856655919644903623367, −7.24535927293442903322178009050, −6.65701570458321410416215636328, −5.66867324082216880767734162525, −5.10630110368576891593301625444, −4.33107301143299464591745424598, −3.55614424485084537046742827374, −3.24764013580982813222844224424, −2.32831049031147362564389318057, −0.903489459514833841238851719565,
0.903489459514833841238851719565, 2.32831049031147362564389318057, 3.24764013580982813222844224424, 3.55614424485084537046742827374, 4.33107301143299464591745424598, 5.10630110368576891593301625444, 5.66867324082216880767734162525, 6.65701570458321410416215636328, 7.24535927293442903322178009050, 7.967064882856655919644903623367
