| L(s) = 1 | + 2-s + 3.01·3-s + 4-s − 4.33·5-s + 3.01·6-s + 1.78·7-s + 8-s + 6.06·9-s − 4.33·10-s − 11-s + 3.01·12-s − 4.54·13-s + 1.78·14-s − 13.0·15-s + 16-s + 5.27·17-s + 6.06·18-s − 4.33·20-s + 5.36·21-s − 22-s − 7.43·23-s + 3.01·24-s + 13.7·25-s − 4.54·26-s + 9.24·27-s + 1.78·28-s + 7.76·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 0.5·4-s − 1.93·5-s + 1.22·6-s + 0.673·7-s + 0.353·8-s + 2.02·9-s − 1.36·10-s − 0.301·11-s + 0.869·12-s − 1.25·13-s + 0.475·14-s − 3.36·15-s + 0.250·16-s + 1.27·17-s + 1.43·18-s − 0.968·20-s + 1.17·21-s − 0.213·22-s − 1.55·23-s + 0.614·24-s + 2.75·25-s − 0.890·26-s + 1.77·27-s + 0.336·28-s + 1.44·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.802980028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.802980028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 + 4.33T + 5T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 - 8.58T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 5.09T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 - 9.86T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 + 2.16T + 71T^{2} \) |
| 73 | \( 1 - 5.98T + 73T^{2} \) |
| 79 | \( 1 + 4.97T + 79T^{2} \) |
| 83 | \( 1 - 6.85T + 83T^{2} \) |
| 89 | \( 1 + 8.73T + 89T^{2} \) |
| 97 | \( 1 - 8.23T + 97T^{2} \) |
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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
−7.74708860636286894896364107511, −7.56896810420737769172758397790, −6.82920958849817997011162891179, −5.51362185517501393642798690913, −4.58438556646495468958748851593, −4.18209485719905658524682287412, −3.58085046628429308549693042594, −2.81648992518530229275094781224, −2.26210369521906263900859175004, −0.903359005611477236711672604714,
0.903359005611477236711672604714, 2.26210369521906263900859175004, 2.81648992518530229275094781224, 3.58085046628429308549693042594, 4.18209485719905658524682287412, 4.58438556646495468958748851593, 5.51362185517501393642798690913, 6.82920958849817997011162891179, 7.56896810420737769172758397790, 7.74708860636286894896364107511
