| L(s) = 1 | + 4.65·2-s + 3·3-s + 13.6·4-s + 13.9·6-s − 26.0·7-s + 26.3·8-s + 9·9-s + 2.51·11-s + 40.9·12-s + 39.6·13-s − 121.·14-s + 13.3·16-s − 84.9·17-s + 41.8·18-s − 142.·19-s − 78.1·21-s + 11.7·22-s + 157.·23-s + 79.0·24-s + 184.·26-s + 27·27-s − 355.·28-s − 100.·29-s − 173.·31-s − 148.·32-s + 7.54·33-s − 395.·34-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 0.577·3-s + 1.70·4-s + 0.950·6-s − 1.40·7-s + 1.16·8-s + 0.333·9-s + 0.0689·11-s + 0.985·12-s + 0.845·13-s − 2.31·14-s + 0.208·16-s − 1.21·17-s + 0.548·18-s − 1.72·19-s − 0.811·21-s + 0.113·22-s + 1.42·23-s + 0.672·24-s + 1.39·26-s + 0.192·27-s − 2.40·28-s − 0.641·29-s − 1.00·31-s − 0.821·32-s + 0.0398·33-s − 1.99·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 4.65T + 8T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 11 | \( 1 - 2.51T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 59.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 109.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 58.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 390.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 351.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 170.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 932.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 653.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 641.T + 9.12e5T^{2} \) |
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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
−8.724451094179160907171617498046, −7.23104312567402399341193333577, −6.60373742781246925817454764734, −6.15651808880359149546002897461, −5.09660994272141721668992819635, −4.12926651479453960846246509850, −3.56910716372289644148438107744, −2.79989897771307039275171772670, −1.88125335525693691736704703775, 0,
1.88125335525693691736704703775, 2.79989897771307039275171772670, 3.56910716372289644148438107744, 4.12926651479453960846246509850, 5.09660994272141721668992819635, 6.15651808880359149546002897461, 6.60373742781246925817454764734, 7.23104312567402399341193333577, 8.724451094179160907171617498046
