| L(s) = 1 | − 2.44·2-s − 8.05·3-s − 2.02·4-s − 13.2·5-s + 19.6·6-s + 1.50·7-s + 24.5·8-s + 37.8·9-s + 32.5·10-s − 19.9·11-s + 16.2·12-s − 49.4·13-s − 3.68·14-s + 107.·15-s − 43.7·16-s − 96.4·17-s − 92.6·18-s + 105.·19-s + 26.8·20-s − 12.1·21-s + 48.8·22-s − 171.·23-s − 197.·24-s + 51.7·25-s + 120.·26-s − 87.7·27-s − 3.04·28-s + ⋯ |
| L(s) = 1 | − 0.864·2-s − 1.55·3-s − 0.252·4-s − 1.18·5-s + 1.34·6-s + 0.0813·7-s + 1.08·8-s + 1.40·9-s + 1.02·10-s − 0.547·11-s + 0.392·12-s − 1.05·13-s − 0.0703·14-s + 1.84·15-s − 0.683·16-s − 1.37·17-s − 1.21·18-s + 1.26·19-s + 0.300·20-s − 0.126·21-s + 0.473·22-s − 1.55·23-s − 1.67·24-s + 0.414·25-s + 0.911·26-s − 0.625·27-s − 0.0205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 8T^{2} \) |
| 3 | \( 1 + 8.05T + 27T^{2} \) |
| 5 | \( 1 + 13.2T + 125T^{2} \) |
| 7 | \( 1 - 1.50T + 343T^{2} \) |
| 11 | \( 1 + 19.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 407.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 482.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 698.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 330.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 307.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 919.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 682.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 411.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 764.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.352317607915151994010392268628, −7.65473950219022785148093507250, −7.18324514105315928771872020089, −6.14127456496762079513268779978, −5.05619397914357785572760954510, −4.65393607437575424064129795698, −3.69742332234260261475127449524, −1.97858094120104937369889881438, −0.56039267969637081043212023375, 0,
0.56039267969637081043212023375, 1.97858094120104937369889881438, 3.69742332234260261475127449524, 4.65393607437575424064129795698, 5.05619397914357785572760954510, 6.14127456496762079513268779978, 7.18324514105315928771872020089, 7.65473950219022785148093507250, 8.352317607915151994010392268628
