L(s) = 1 | + 1.55·2-s + 3·3-s − 5.57·4-s + 9.75·5-s + 4.66·6-s − 7·7-s − 21.1·8-s + 9·9-s + 15.1·10-s − 48.6·11-s − 16.7·12-s + 21.0·13-s − 10.8·14-s + 29.2·15-s + 11.7·16-s − 48.9·17-s + 14.0·18-s − 35.9·19-s − 54.4·20-s − 21·21-s − 75.7·22-s − 23·23-s − 63.3·24-s − 29.8·25-s + 32.6·26-s + 27·27-s + 39.0·28-s + ⋯ |
L(s) = 1 | + 0.550·2-s + 0.577·3-s − 0.697·4-s + 0.872·5-s + 0.317·6-s − 0.377·7-s − 0.933·8-s + 0.333·9-s + 0.479·10-s − 1.33·11-s − 0.402·12-s + 0.448·13-s − 0.207·14-s + 0.503·15-s + 0.183·16-s − 0.698·17-s + 0.183·18-s − 0.434·19-s − 0.608·20-s − 0.218·21-s − 0.733·22-s − 0.208·23-s − 0.539·24-s − 0.238·25-s + 0.246·26-s + 0.192·27-s + 0.263·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
good | 2 | \( 1 - 1.55T + 8T^{2} \) |
| 5 | \( 1 - 9.75T + 125T^{2} \) |
| 11 | \( 1 + 48.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 89.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 461.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 775.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 304.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 192.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 19.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 757.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 148.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 41.6T + 9.12e5T^{2} \) |
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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
−10.03629838532778585714031700240, −9.184462122179408590608198851514, −8.533132065291115411508675987693, −7.40372806838696290055718392540, −6.08512196606651966832478885690, −5.38676096228515099189684961170, −4.25429012729173676755101026487, −3.15470112844774025343378443854, −2.02968768820791289761346491683, 0,
2.02968768820791289761346491683, 3.15470112844774025343378443854, 4.25429012729173676755101026487, 5.38676096228515099189684961170, 6.08512196606651966832478885690, 7.40372806838696290055718392540, 8.533132065291115411508675987693, 9.184462122179408590608198851514, 10.03629838532778585714031700240