| L(s) = 1 | − 1.24e11·2-s + 1.35e17·3-s − 2.21e22·4-s + 1.80e26·5-s − 1.68e28·6-s + 2.76e31·7-s + 7.49e33·8-s − 5.90e35·9-s − 2.26e37·10-s + 7.60e38·11-s − 2.99e39·12-s + 2.57e41·13-s − 3.45e42·14-s + 2.44e43·15-s − 9.94e43·16-s − 1.39e46·17-s + 7.37e46·18-s − 1.16e48·19-s − 4.00e48·20-s + 3.73e48·21-s − 9.50e49·22-s + 9.81e50·23-s + 1.01e51·24-s + 6.26e51·25-s − 3.22e52·26-s − 1.61e53·27-s − 6.11e53·28-s + ⋯ |
| L(s) = 1 | − 0.643·2-s + 0.173·3-s − 0.586·4-s + 1.11·5-s − 0.111·6-s + 0.562·7-s + 1.02·8-s − 0.969·9-s − 0.715·10-s + 0.674·11-s − 0.101·12-s + 0.434·13-s − 0.361·14-s + 0.192·15-s − 0.0697·16-s − 1.00·17-s + 0.623·18-s − 1.29·19-s − 0.652·20-s + 0.0974·21-s − 0.433·22-s + 0.845·23-s + 0.176·24-s + 0.236·25-s − 0.279·26-s − 0.341·27-s − 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(76-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+75/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(1.610368071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610368071\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.24e11T + 3.77e22T^{2} \) |
| 3 | \( 1 - 1.35e17T + 6.08e35T^{2} \) |
| 5 | \( 1 - 1.80e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 2.76e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 7.60e38T + 1.27e78T^{2} \) |
| 13 | \( 1 - 2.57e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 1.39e46T + 1.92e92T^{2} \) |
| 19 | \( 1 + 1.16e48T + 8.06e95T^{2} \) |
| 23 | \( 1 - 9.81e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 3.47e54T + 4.78e109T^{2} \) |
| 31 | \( 1 - 1.19e56T + 7.11e111T^{2} \) |
| 37 | \( 1 - 1.03e59T + 4.12e117T^{2} \) |
| 41 | \( 1 - 4.07e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 2.20e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 4.37e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 7.68e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 3.79e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 5.02e66T + 7.93e133T^{2} \) |
| 67 | \( 1 - 3.39e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 3.17e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 7.11e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 9.47e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 1.43e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 9.94e72T + 1.60e146T^{2} \) |
| 97 | \( 1 + 7.42e73T + 1.01e149T^{2} \) |
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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
−17.08849976426933116417692103830, −14.47524515780867517424241644882, −13.33854408255827387586920177614, −10.95628709701300446889046117978, −9.340907332054442788083289369497, −8.375895378057554883273929259773, −6.16233581584241431351625569540, −4.48881368106090201687271821890, −2.27943924184356855781281266347, −0.874762885394372842178720411755,
0.874762885394372842178720411755, 2.27943924184356855781281266347, 4.48881368106090201687271821890, 6.16233581584241431351625569540, 8.375895378057554883273929259773, 9.340907332054442788083289369497, 10.95628709701300446889046117978, 13.33854408255827387586920177614, 14.47524515780867517424241644882, 17.08849976426933116417692103830
