| L(s) = 1 | − 2.62·2-s + 1.98·3-s + 4.87·4-s − 3.39·5-s − 5.21·6-s − 2.62·7-s − 7.52·8-s + 0.954·9-s + 8.90·10-s − 11-s + 9.68·12-s − 4.77·13-s + 6.88·14-s − 6.75·15-s + 9.98·16-s + 6.37·17-s − 2.50·18-s + 1.24·19-s − 16.5·20-s − 5.22·21-s + 2.62·22-s + 0.341·23-s − 14.9·24-s + 6.55·25-s + 12.5·26-s − 4.06·27-s − 12.7·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.14·3-s + 2.43·4-s − 1.52·5-s − 2.12·6-s − 0.992·7-s − 2.66·8-s + 0.318·9-s + 2.81·10-s − 0.301·11-s + 2.79·12-s − 1.32·13-s + 1.83·14-s − 1.74·15-s + 2.49·16-s + 1.54·17-s − 0.589·18-s + 0.286·19-s − 3.70·20-s − 1.13·21-s + 0.558·22-s + 0.0711·23-s − 3.05·24-s + 1.31·25-s + 2.45·26-s − 0.782·27-s − 2.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 1.98T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 13 | \( 1 + 4.77T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 0.341T + 23T^{2} \) |
| 29 | \( 1 - 7.39T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 1.40T + 97T^{2} \) |
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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
−7.77800888759826732317984440515, −7.56186207842583780249525674098, −6.88810633411806137121979795475, −5.96194322988708881719922313520, −4.67805983579963708573021510837, −3.44438657309698912603181964097, −3.04018739037510358108369689163, −2.38480272065560180555712930987, −0.950392526062304158363768943038, 0,
0.950392526062304158363768943038, 2.38480272065560180555712930987, 3.04018739037510358108369689163, 3.44438657309698912603181964097, 4.67805983579963708573021510837, 5.96194322988708881719922313520, 6.88810633411806137121979795475, 7.56186207842583780249525674098, 7.77800888759826732317984440515
