| L(s) = 1 | − 1.09e12·2-s + 2.69e19·3-s + 1.20e24·4-s + 1.65e28·5-s − 2.96e31·6-s + 2.15e33·7-s − 1.32e36·8-s + 2.82e38·9-s − 1.82e40·10-s − 3.85e40·11-s + 3.25e43·12-s − 8.46e43·13-s − 2.36e45·14-s + 4.46e47·15-s + 1.46e48·16-s − 7.68e49·17-s − 3.10e50·18-s − 7.37e51·19-s + 2.00e52·20-s + 5.80e52·21-s + 4.23e52·22-s − 1.05e55·23-s − 3.58e55·24-s − 1.39e56·25-s + 9.31e55·26-s − 4.34e57·27-s + 2.60e57·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.27·3-s + 0.5·4-s + 0.814·5-s − 0.904·6-s + 0.127·7-s − 0.353·8-s + 0.636·9-s − 0.575·10-s − 0.0256·11-s + 0.639·12-s − 0.0650·13-s − 0.0904·14-s + 1.04·15-s + 0.250·16-s − 1.12·17-s − 0.450·18-s − 1.19·19-s + 0.407·20-s + 0.163·21-s + 0.0181·22-s − 0.745·23-s − 0.452·24-s − 0.337·25-s + 0.0459·26-s − 0.465·27-s + 0.0639·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(82-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+81/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(41)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{83}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.09e12T \) |
| good | 3 | \( 1 - 2.69e19T + 4.43e38T^{2} \) |
| 5 | \( 1 - 1.65e28T + 4.13e56T^{2} \) |
| 7 | \( 1 - 2.15e33T + 2.83e68T^{2} \) |
| 11 | \( 1 + 3.85e40T + 2.25e84T^{2} \) |
| 13 | \( 1 + 8.46e43T + 1.69e90T^{2} \) |
| 17 | \( 1 + 7.68e49T + 4.63e99T^{2} \) |
| 19 | \( 1 + 7.37e51T + 3.79e103T^{2} \) |
| 23 | \( 1 + 1.05e55T + 1.99e110T^{2} \) |
| 29 | \( 1 + 1.79e59T + 2.84e118T^{2} \) |
| 31 | \( 1 + 4.86e60T + 6.31e120T^{2} \) |
| 37 | \( 1 - 2.30e63T + 1.05e127T^{2} \) |
| 41 | \( 1 + 1.35e65T + 4.32e130T^{2} \) |
| 43 | \( 1 - 2.10e66T + 2.04e132T^{2} \) |
| 47 | \( 1 - 1.03e68T + 2.75e135T^{2} \) |
| 53 | \( 1 - 2.99e69T + 4.63e139T^{2} \) |
| 59 | \( 1 + 5.68e71T + 2.74e143T^{2} \) |
| 61 | \( 1 - 5.44e71T + 4.08e144T^{2} \) |
| 67 | \( 1 + 5.75e73T + 8.16e147T^{2} \) |
| 71 | \( 1 - 4.64e74T + 8.95e149T^{2} \) |
| 73 | \( 1 - 4.70e75T + 8.49e150T^{2} \) |
| 79 | \( 1 + 4.20e76T + 5.10e153T^{2} \) |
| 83 | \( 1 + 7.46e77T + 2.78e155T^{2} \) |
| 89 | \( 1 - 5.59e78T + 7.95e157T^{2} \) |
| 97 | \( 1 + 1.28e80T + 8.48e160T^{2} \) |
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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
−12.96446727965251858602492250770, −10.91540996963695945676992130567, −9.494518375178867106013539250488, −8.723154818811695291405901533350, −7.47186979821420211745410067144, −5.94217770458994949981878820727, −3.93042105477405562264675147715, −2.36424398822134285468909051906, −1.85240261465840691084710946396, 0,
1.85240261465840691084710946396, 2.36424398822134285468909051906, 3.93042105477405562264675147715, 5.94217770458994949981878820727, 7.47186979821420211745410067144, 8.723154818811695291405901533350, 9.494518375178867106013539250488, 10.91540996963695945676992130567, 12.96446727965251858602492250770
