| L(s) = 1 | + 2.70·2-s + 3·3-s − 0.701·4-s + 5·5-s + 8.10·6-s − 10.1·7-s − 23.5·8-s + 9·9-s + 13.5·10-s − 2.10·12-s − 26.7·13-s − 27.2·14-s + 15·15-s − 57.8·16-s + 36.1·17-s + 24.3·18-s + 66.3·19-s − 3.50·20-s − 30.3·21-s + 70.7·23-s − 70.5·24-s + 25·25-s − 72.1·26-s + 27·27-s + 7.08·28-s + 41.0·29-s + 40.5·30-s + ⋯ |
| L(s) = 1 | + 0.955·2-s + 0.577·3-s − 0.0876·4-s + 0.447·5-s + 0.551·6-s − 0.545·7-s − 1.03·8-s + 0.333·9-s + 0.427·10-s − 0.0506·12-s − 0.569·13-s − 0.521·14-s + 0.258·15-s − 0.904·16-s + 0.516·17-s + 0.318·18-s + 0.800·19-s − 0.0392·20-s − 0.315·21-s + 0.641·23-s − 0.599·24-s + 0.200·25-s − 0.544·26-s + 0.192·27-s + 0.0478·28-s + 0.263·29-s + 0.246·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.70T + 8T^{2} \) |
| 7 | \( 1 + 10.1T + 343T^{2} \) |
| 13 | \( 1 + 26.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 66.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 24.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 367.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 79.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 99.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 460.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 424.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 263.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 117.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 535.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 823.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 128.T + 9.12e5T^{2} \) |
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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
−8.656113016066284948045230769136, −7.62664622133695071014941374278, −6.79645214995982563222528638763, −5.93161425243261864579820129592, −5.14888508436634696096398563784, −4.41942383187225474871716831181, −3.24721496128905346592216008810, −2.94038978992505287738921907214, −1.53063271734471398742545799061, 0,
1.53063271734471398742545799061, 2.94038978992505287738921907214, 3.24721496128905346592216008810, 4.41942383187225474871716831181, 5.14888508436634696096398563784, 5.93161425243261864579820129592, 6.79645214995982563222528638763, 7.62664622133695071014941374278, 8.656113016066284948045230769136
