L(s) = 1 | − 2.59·2-s + 3.24·3-s + 4.72·4-s − 1.12·5-s − 8.40·6-s − 0.119·7-s − 7.05·8-s + 7.51·9-s + 2.91·10-s − 1.92·11-s + 15.3·12-s + 3.49·13-s + 0.309·14-s − 3.64·15-s + 8.85·16-s + 0.738·17-s − 19.4·18-s − 7.27·19-s − 5.30·20-s − 0.386·21-s + 4.98·22-s + 7.57·23-s − 22.8·24-s − 3.73·25-s − 9.07·26-s + 14.6·27-s − 0.563·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 1.87·3-s + 2.36·4-s − 0.502·5-s − 3.43·6-s − 0.0450·7-s − 2.49·8-s + 2.50·9-s + 0.921·10-s − 0.579·11-s + 4.41·12-s + 0.970·13-s + 0.0826·14-s − 0.941·15-s + 2.21·16-s + 0.179·17-s − 4.59·18-s − 1.66·19-s − 1.18·20-s − 0.0843·21-s + 1.06·22-s + 1.57·23-s − 4.67·24-s − 0.747·25-s − 1.77·26-s + 2.81·27-s − 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 0.119T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 - 0.738T + 17T^{2} \) |
| 19 | \( 1 + 7.27T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 0.406T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 8.19T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 3.82T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 + 4.08T + 79T^{2} \) |
| 83 | \( 1 - 3.38T + 83T^{2} \) |
| 89 | \( 1 - 2.04T + 89T^{2} \) |
| 97 | \( 1 + 1.22T + 97T^{2} \) |
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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
−7.950589807777121024327325038045, −7.41759611551183669848270844847, −6.93110816912913928327839393837, −5.98836740980496589331718261154, −4.54645199301175423456994272894, −3.50025891751203679552727405507, −3.06203480485483695308012826134, −1.97855599679975364759655057507, −1.55090095468504953602593138910, 0,
1.55090095468504953602593138910, 1.97855599679975364759655057507, 3.06203480485483695308012826134, 3.50025891751203679552727405507, 4.54645199301175423456994272894, 5.98836740980496589331718261154, 6.93110816912913928327839393837, 7.41759611551183669848270844847, 7.950589807777121024327325038045