| L(s) = 1 | − 2.23·2-s + 3.00·4-s + 2·5-s − 7-s − 2.23·8-s − 4.47·10-s + 1.23·13-s + 2.23·14-s − 0.999·16-s + 1.23·17-s + 2.47·19-s + 6.00·20-s + 6.47·23-s − 25-s − 2.76·26-s − 3.00·28-s − 0.472·29-s − 7.23·31-s + 6.70·32-s − 2.76·34-s − 2·35-s + 0.472·37-s − 5.52·38-s − 4.47·40-s − 6.76·41-s − 8·43-s − 14.4·46-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + 0.894·5-s − 0.377·7-s − 0.790·8-s − 1.41·10-s + 0.342·13-s + 0.597·14-s − 0.249·16-s + 0.299·17-s + 0.567·19-s + 1.34·20-s + 1.34·23-s − 0.200·25-s − 0.542·26-s − 0.566·28-s − 0.0876·29-s − 1.29·31-s + 1.18·32-s − 0.474·34-s − 0.338·35-s + 0.0776·37-s − 0.896·38-s − 0.707·40-s − 1.05·41-s − 1.21·43-s − 2.13·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−7.72862752904650541078578623544, −6.84178033878918086473019447893, −6.54585758853931192056088207643, −5.55162246720538219839015970625, −4.95672095849414624037505487683, −3.63534980479515226249206214833, −2.84620157227462634890446400260, −1.83697598738312793209164987775, −1.24665757722581832356087043882, 0,
1.24665757722581832356087043882, 1.83697598738312793209164987775, 2.84620157227462634890446400260, 3.63534980479515226249206214833, 4.95672095849414624037505487683, 5.55162246720538219839015970625, 6.54585758853931192056088207643, 6.84178033878918086473019447893, 7.72862752904650541078578623544
