| L(s) = 1 | − 1.28e10·2-s − 9.28e21·4-s − 4.05e25·5-s − 7.12e30·7-s + 2.39e32·8-s + 5.19e35·10-s − 5.42e37·11-s + 7.16e40·13-s + 9.12e40·14-s + 8.45e43·16-s − 1.30e44·17-s − 2.65e46·19-s + 3.76e47·20-s + 6.94e47·22-s + 6.34e49·23-s + 5.89e50·25-s − 9.17e50·26-s + 6.61e52·28-s − 8.00e52·29-s − 3.53e54·31-s − 3.34e54·32-s + 1.67e54·34-s + 2.89e56·35-s + 1.61e56·37-s + 3.40e56·38-s − 9.73e57·40-s − 6.62e58·41-s + ⋯ |
| L(s) = 1 | − 0.131·2-s − 0.982·4-s − 1.24·5-s − 1.01·7-s + 0.261·8-s + 0.164·10-s − 0.529·11-s + 1.57·13-s + 0.133·14-s + 0.948·16-s − 0.160·17-s − 0.561·19-s + 1.22·20-s + 0.0697·22-s + 1.25·23-s + 0.556·25-s − 0.206·26-s + 0.997·28-s − 0.335·29-s − 1.29·31-s − 0.386·32-s + 0.0211·34-s + 1.26·35-s + 0.0928·37-s + 0.0740·38-s − 0.325·40-s − 0.900·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(\approx\) |
\(0.1660589948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1660589948\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.28e10T + 9.44e21T^{2} \) |
| 5 | \( 1 + 4.05e25T + 1.05e51T^{2} \) |
| 7 | \( 1 + 7.12e30T + 4.92e61T^{2} \) |
| 11 | \( 1 + 5.42e37T + 1.05e76T^{2} \) |
| 13 | \( 1 - 7.16e40T + 2.07e81T^{2} \) |
| 17 | \( 1 + 1.30e44T + 6.64e89T^{2} \) |
| 19 | \( 1 + 2.65e46T + 2.23e93T^{2} \) |
| 23 | \( 1 - 6.34e49T + 2.54e99T^{2} \) |
| 29 | \( 1 + 8.00e52T + 5.68e106T^{2} \) |
| 31 | \( 1 + 3.53e54T + 7.40e108T^{2} \) |
| 37 | \( 1 - 1.61e56T + 3.01e114T^{2} \) |
| 41 | \( 1 + 6.62e58T + 5.41e117T^{2} \) |
| 43 | \( 1 - 5.13e57T + 1.75e119T^{2} \) |
| 47 | \( 1 + 1.87e61T + 1.15e122T^{2} \) |
| 53 | \( 1 - 2.36e62T + 7.44e125T^{2} \) |
| 59 | \( 1 - 2.19e64T + 1.87e129T^{2} \) |
| 61 | \( 1 + 1.30e65T + 2.13e130T^{2} \) |
| 67 | \( 1 + 5.38e66T + 2.01e133T^{2} \) |
| 71 | \( 1 + 3.88e67T + 1.38e135T^{2} \) |
| 73 | \( 1 + 9.28e67T + 1.05e136T^{2} \) |
| 79 | \( 1 + 4.44e68T + 3.36e138T^{2} \) |
| 83 | \( 1 + 9.20e69T + 1.23e140T^{2} \) |
| 89 | \( 1 - 3.27e70T + 2.02e142T^{2} \) |
| 97 | \( 1 + 2.09e72T + 1.08e145T^{2} \) |
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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
−10.39373533160648450068714938366, −9.077696306700134523580540996711, −8.383956338901904903730156883374, −7.28541291872785138937718589066, −6.02639315406019977169084998473, −4.75502574496530545684349066472, −3.71672667187815653337027533302, −3.21555813335174562667940214041, −1.34595218439293030615013927962, −0.17530830120959532772922126499,
0.17530830120959532772922126499, 1.34595218439293030615013927962, 3.21555813335174562667940214041, 3.71672667187815653337027533302, 4.75502574496530545684349066472, 6.02639315406019977169084998473, 7.28541291872785138937718589066, 8.383956338901904903730156883374, 9.077696306700134523580540996711, 10.39373533160648450068714938366
