L(s) = 1 | − 2.57·2-s + 1.64·3-s + 4.63·4-s − 3.44·5-s − 4.24·6-s − 0.596·7-s − 6.77·8-s − 0.283·9-s + 8.88·10-s + 5.70·11-s + 7.63·12-s + 1.47·13-s + 1.53·14-s − 5.68·15-s + 8.18·16-s + 1.39·17-s + 0.729·18-s − 4.14·19-s − 15.9·20-s − 0.982·21-s − 14.6·22-s − 2.33·23-s − 11.1·24-s + 6.89·25-s − 3.79·26-s − 5.41·27-s − 2.76·28-s + ⋯ |
L(s) = 1 | − 1.82·2-s + 0.951·3-s + 2.31·4-s − 1.54·5-s − 1.73·6-s − 0.225·7-s − 2.39·8-s − 0.0944·9-s + 2.80·10-s + 1.72·11-s + 2.20·12-s + 0.408·13-s + 0.410·14-s − 1.46·15-s + 2.04·16-s + 0.339·17-s + 0.171·18-s − 0.951·19-s − 3.57·20-s − 0.214·21-s − 3.13·22-s − 0.487·23-s − 2.27·24-s + 1.37·25-s − 0.744·26-s − 1.04·27-s − 0.521·28-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 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 + 3.44T + 5T^{2} \) |
| 7 | \( 1 + 0.596T + 7T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 1.39T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 + 0.594T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 + 0.691T + 41T^{2} \) |
| 43 | \( 1 + 8.85T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 + 1.09T + 61T^{2} \) |
| 67 | \( 1 - 5.66T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + 7.01T + 73T^{2} \) |
| 79 | \( 1 - 5.98T + 79T^{2} \) |
| 83 | \( 1 - 0.0878T + 83T^{2} \) |
| 89 | \( 1 - 0.350T + 89T^{2} \) |
| 97 | \( 1 - 6.42T + 97T^{2} \) |
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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
−8.148145793847615602479954399251, −7.32354169742938758817299206210, −6.68275922984199098538700783224, −6.15638712108036052953402232644, −4.50642348990677961457932061206, −3.64122348313235696422119563749, −3.17215613188975182844394495745, −2.06277264934406480300653545891, −1.10674181216466080983581802454, 0,
1.10674181216466080983581802454, 2.06277264934406480300653545891, 3.17215613188975182844394495745, 3.64122348313235696422119563749, 4.50642348990677961457932061206, 6.15638712108036052953402232644, 6.68275922984199098538700783224, 7.32354169742938758817299206210, 8.148145793847615602479954399251