| L(s) = 1 | + 6.87e10·2-s + 4.40e17·3-s + 4.72e21·4-s − 6.01e25·5-s + 3.02e28·6-s + 1.36e30·7-s + 3.24e32·8-s + 1.26e35·9-s − 4.13e36·10-s + 1.17e38·11-s + 2.08e39·12-s − 4.64e40·13-s + 9.39e40·14-s − 2.65e43·15-s + 2.23e43·16-s + 7.41e44·17-s + 8.70e45·18-s + 3.86e46·19-s − 2.84e47·20-s + 6.02e47·21-s + 8.06e48·22-s + 5.58e49·23-s + 1.43e50·24-s + 2.56e51·25-s − 3.18e51·26-s + 2.60e52·27-s + 6.45e51·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s − 1.84·5-s + 1.19·6-s + 0.194·7-s + 0.353·8-s + 1.87·9-s − 1.30·10-s + 1.14·11-s + 0.847·12-s − 1.01·13-s + 0.137·14-s − 3.13·15-s + 0.250·16-s + 0.909·17-s + 1.32·18-s + 0.818·19-s − 0.924·20-s + 0.330·21-s + 0.809·22-s + 1.10·23-s + 0.599·24-s + 2.41·25-s − 0.719·26-s + 1.48·27-s + 0.0974·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(\approx\) |
\(5.323029320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.323029320\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.87e10T \) |
| good | 3 | \( 1 - 4.40e17T + 6.75e34T^{2} \) |
| 5 | \( 1 + 6.01e25T + 1.05e51T^{2} \) |
| 7 | \( 1 - 1.36e30T + 4.92e61T^{2} \) |
| 11 | \( 1 - 1.17e38T + 1.05e76T^{2} \) |
| 13 | \( 1 + 4.64e40T + 2.07e81T^{2} \) |
| 17 | \( 1 - 7.41e44T + 6.64e89T^{2} \) |
| 19 | \( 1 - 3.86e46T + 2.23e93T^{2} \) |
| 23 | \( 1 - 5.58e49T + 2.54e99T^{2} \) |
| 29 | \( 1 + 1.18e53T + 5.68e106T^{2} \) |
| 31 | \( 1 - 2.47e52T + 7.40e108T^{2} \) |
| 37 | \( 1 - 2.03e57T + 3.01e114T^{2} \) |
| 41 | \( 1 - 1.41e59T + 5.41e117T^{2} \) |
| 43 | \( 1 - 2.36e59T + 1.75e119T^{2} \) |
| 47 | \( 1 + 7.61e60T + 1.15e122T^{2} \) |
| 53 | \( 1 + 6.20e62T + 7.44e125T^{2} \) |
| 59 | \( 1 - 4.74e64T + 1.87e129T^{2} \) |
| 61 | \( 1 - 6.52e64T + 2.13e130T^{2} \) |
| 67 | \( 1 + 7.25e66T + 2.01e133T^{2} \) |
| 71 | \( 1 + 2.81e67T + 1.38e135T^{2} \) |
| 73 | \( 1 - 5.22e67T + 1.05e136T^{2} \) |
| 79 | \( 1 - 6.42e68T + 3.36e138T^{2} \) |
| 83 | \( 1 - 1.58e69T + 1.23e140T^{2} \) |
| 89 | \( 1 - 4.25e69T + 2.02e142T^{2} \) |
| 97 | \( 1 + 2.34e72T + 1.08e145T^{2} \) |
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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
−14.49251988542826804073083791635, −12.67129273583070519644680532602, −11.50091278316374071887272845431, −9.321311394281575963670467388630, −7.889789903966694908870594716120, −7.24522213843305328290322266895, −4.54681436659644297667605696538, −3.63306938722294399388880299348, −2.81901243447341161578037480170, −1.08893243391484536301450751520,
1.08893243391484536301450751520, 2.81901243447341161578037480170, 3.63306938722294399388880299348, 4.54681436659644297667605696538, 7.24522213843305328290322266895, 7.889789903966694908870594716120, 9.321311394281575963670467388630, 11.50091278316374071887272845431, 12.67129273583070519644680532602, 14.49251988542826804073083791635
