| L(s) = 1 | + 3.60·2-s + 7.89·3-s + 5.03·4-s + 28.4·6-s − 27.9·7-s − 10.7·8-s + 35.3·9-s − 41.0·11-s + 39.7·12-s + 76.6·13-s − 101.·14-s − 78.9·16-s + 21.4·17-s + 127.·18-s + 38.3·19-s − 220.·21-s − 148.·22-s − 22.8·23-s − 84.6·24-s + 276.·26-s + 65.6·27-s − 140.·28-s − 256.·29-s − 37.0·31-s − 199.·32-s − 323.·33-s + 77.2·34-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 1.51·3-s + 0.628·4-s + 1.93·6-s − 1.51·7-s − 0.473·8-s + 1.30·9-s − 1.12·11-s + 0.955·12-s + 1.63·13-s − 1.92·14-s − 1.23·16-s + 0.305·17-s + 1.66·18-s + 0.462·19-s − 2.29·21-s − 1.43·22-s − 0.206·23-s − 0.719·24-s + 2.08·26-s + 0.467·27-s − 0.950·28-s − 1.64·29-s − 0.214·31-s − 1.10·32-s − 1.70·33-s + 0.389·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 71 | \( 1 - 71T \) |
| good | 2 | \( 1 - 3.60T + 8T^{2} \) |
| 3 | \( 1 - 7.89T + 27T^{2} \) |
| 7 | \( 1 + 27.9T + 343T^{2} \) |
| 11 | \( 1 + 41.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 37.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 141.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 45.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 340.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 31.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 67.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 419.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 94.3T + 3.00e5T^{2} \) |
| 73 | \( 1 + 313.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 645.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 412.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 620.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.39e3T + 9.12e5T^{2} \) |
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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
−8.595957743681044515992820994155, −7.75974413416585671561604582955, −6.80989680518522965314929091388, −6.00299278615566094739809098547, −5.24234023086363677388450519730, −3.94192657962933285139881688232, −3.35142624908636143841982956484, −3.05395033695872706470046170898, −1.87940708749267780343723254904, 0,
1.87940708749267780343723254904, 3.05395033695872706470046170898, 3.35142624908636143841982956484, 3.94192657962933285139881688232, 5.24234023086363677388450519730, 6.00299278615566094739809098547, 6.80989680518522965314929091388, 7.75974413416585671561604582955, 8.595957743681044515992820994155
