| L(s) = 1 | − 1.12·2-s + 3-s − 0.727·4-s + 5-s − 1.12·6-s − 4.89·7-s + 3.07·8-s + 9-s − 1.12·10-s + 4.95·11-s − 0.727·12-s + 2.22·13-s + 5.52·14-s + 15-s − 2.01·16-s − 2.48·17-s − 1.12·18-s − 7.75·19-s − 0.727·20-s − 4.89·21-s − 5.59·22-s − 0.154·23-s + 3.07·24-s + 25-s − 2.51·26-s + 27-s + 3.55·28-s + ⋯ |
| L(s) = 1 | − 0.797·2-s + 0.577·3-s − 0.363·4-s + 0.447·5-s − 0.460·6-s − 1.84·7-s + 1.08·8-s + 0.333·9-s − 0.356·10-s + 1.49·11-s − 0.209·12-s + 0.617·13-s + 1.47·14-s + 0.258·15-s − 0.504·16-s − 0.601·17-s − 0.265·18-s − 1.77·19-s − 0.162·20-s − 1.06·21-s − 1.19·22-s − 0.0322·23-s + 0.628·24-s + 0.200·25-s − 0.493·26-s + 0.192·27-s + 0.672·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 + 0.154T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.72T + 41T^{2} \) |
| 43 | \( 1 + 1.74T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 - 8.73T + 97T^{2} \) |
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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
−7.949959385492318453830695448224, −6.93352700568510687714901696250, −6.41053660475853549436210848877, −6.01016466810286025895790132258, −4.47793456604547593026598192398, −4.03310014849498971541531135382, −3.22540840424235954842182896372, −2.20259826204299516002097562802, −1.19978761760043757911341104285, 0,
1.19978761760043757911341104285, 2.20259826204299516002097562802, 3.22540840424235954842182896372, 4.03310014849498971541531135382, 4.47793456604547593026598192398, 6.01016466810286025895790132258, 6.41053660475853549436210848877, 6.93352700568510687714901696250, 7.949959385492318453830695448224
