| L(s) = 1 | − 2.95·2-s − 3·3-s + 0.744·4-s − 5·5-s + 8.87·6-s + 13.4·7-s + 21.4·8-s + 9·9-s + 14.7·10-s − 2.23·12-s + 13.8·13-s − 39.7·14-s + 15·15-s − 69.4·16-s + 25.6·17-s − 26.6·18-s + 38.1·19-s − 3.72·20-s − 40.3·21-s − 62.5·23-s − 64.3·24-s + 25·25-s − 40.9·26-s − 27·27-s + 10.0·28-s + 53.6·29-s − 44.3·30-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.577·3-s + 0.0931·4-s − 0.447·5-s + 0.603·6-s + 0.725·7-s + 0.948·8-s + 0.333·9-s + 0.467·10-s − 0.0537·12-s + 0.295·13-s − 0.758·14-s + 0.258·15-s − 1.08·16-s + 0.365·17-s − 0.348·18-s + 0.460·19-s − 0.0416·20-s − 0.418·21-s − 0.567·23-s − 0.547·24-s + 0.200·25-s − 0.308·26-s − 0.192·27-s + 0.0675·28-s + 0.343·29-s − 0.269·30-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 | \( 1 + 2.95T + 8T^{2} \) |
| 7 | \( 1 - 13.4T + 343T^{2} \) |
| 13 | \( 1 - 13.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 53.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 259.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 20.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 697.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 829.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 445.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 718.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 268.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 368.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 296.T + 9.12e5T^{2} \) |
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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
−8.537893284830032877117418102023, −7.72790390840825719547638324670, −7.33519383871179591598072716557, −6.19495247916609157532717042611, −5.20123674532337863404489710197, −4.49173586290607171325257122016, −3.52141438495810549220150678751, −1.91309749019183318357975996535, −1.02879874149294061782111882990, 0,
1.02879874149294061782111882990, 1.91309749019183318357975996535, 3.52141438495810549220150678751, 4.49173586290607171325257122016, 5.20123674532337863404489710197, 6.19495247916609157532717042611, 7.33519383871179591598072716557, 7.72790390840825719547638324670, 8.537893284830032877117418102023
