| L(s) = 1 | − 4.09e3i·2-s − 1.37e6i·3-s − 1.67e7·4-s − 5.64e9·6-s − 3.00e10i·7-s + 6.87e10i·8-s − 1.05e12·9-s + 5.73e12·11-s + 2.31e13i·12-s − 1.07e13i·13-s − 1.23e14·14-s + 2.81e14·16-s − 2.97e15i·17-s + 4.30e15i·18-s + 5.42e15·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.49i·3-s − 0.5·4-s − 1.05·6-s − 0.820i·7-s + 0.353i·8-s − 1.24·9-s + 0.551·11-s + 0.748i·12-s − 0.127i·13-s − 0.580·14-s + 0.250·16-s − 1.23i·17-s + 0.877i·18-s + 0.562·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.6579105756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6579105756\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.37e6iT - 8.47e11T^{2} \) |
| 7 | \( 1 + 3.00e10iT - 1.34e21T^{2} \) |
| 11 | \( 1 - 5.73e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.07e13iT - 7.05e27T^{2} \) |
| 17 | \( 1 + 2.97e15iT - 5.77e30T^{2} \) |
| 19 | \( 1 - 5.42e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.04e17iT - 1.10e34T^{2} \) |
| 29 | \( 1 + 3.09e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.26e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 4.51e19iT - 1.60e39T^{2} \) |
| 41 | \( 1 + 7.56e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.39e20iT - 6.86e40T^{2} \) |
| 47 | \( 1 + 4.67e20iT - 6.34e41T^{2} \) |
| 53 | \( 1 + 1.24e21iT - 1.27e43T^{2} \) |
| 59 | \( 1 - 1.32e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.36e20T + 4.29e44T^{2} \) |
| 67 | \( 1 - 5.36e22iT - 4.48e45T^{2} \) |
| 71 | \( 1 + 2.27e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 2.78e23iT - 3.82e46T^{2} \) |
| 79 | \( 1 + 6.36e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.19e24iT - 9.48e47T^{2} \) |
| 89 | \( 1 - 3.22e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 3.58e24iT - 4.66e49T^{2} \) |
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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
−9.943119823132992242045974515910, −8.696610054118011315564982805106, −7.46285514989263660077472779651, −6.83260690923834460426070622502, −5.41339245283069785351579785328, −3.97423059780014756664365834155, −2.76855168595841354439362799529, −1.70862222870861534323670451947, −0.890390230079325840557456188095, −0.13741689015214069820301177084,
1.70965489106963168748719771595, 3.38904536512428177302423954753, 4.10527584878971583552289137380, 5.31237617009708272333712442605, 5.96989127456670900819980476544, 7.55191980939334204289506396456, 8.952492996312759714923292827002, 9.369700915199391981878560708958, 10.57100009485619627096151815475, 11.67313288828845731021774052976
