| L(s) = 1 | − 2.51·2-s + 3.19·3-s + 4.31·4-s − 5-s − 8.01·6-s + 7-s − 5.82·8-s + 7.17·9-s + 2.51·10-s − 1.97·11-s + 13.7·12-s − 0.775·13-s − 2.51·14-s − 3.19·15-s + 6.01·16-s + 3.70·17-s − 18.0·18-s − 7.35·19-s − 4.31·20-s + 3.19·21-s + 4.97·22-s + 0.639·23-s − 18.5·24-s + 25-s + 1.94·26-s + 13.3·27-s + 4.31·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 1.84·3-s + 2.15·4-s − 0.447·5-s − 3.27·6-s + 0.377·7-s − 2.06·8-s + 2.39·9-s + 0.794·10-s − 0.596·11-s + 3.97·12-s − 0.215·13-s − 0.671·14-s − 0.823·15-s + 1.50·16-s + 0.898·17-s − 4.25·18-s − 1.68·19-s − 0.965·20-s + 0.696·21-s + 1.06·22-s + 0.133·23-s − 3.79·24-s + 0.200·25-s + 0.382·26-s + 2.56·27-s + 0.816·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 3.19T + 3T^{2} \) |
| 11 | \( 1 + 1.97T + 11T^{2} \) |
| 13 | \( 1 + 0.775T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 7.35T + 19T^{2} \) |
| 23 | \( 1 - 0.639T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 0.116T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 3.97T + 59T^{2} \) |
| 61 | \( 1 - 0.734T + 61T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 3.41T + 97T^{2} \) |
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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
−7.72756384246078412459714876304, −7.30192869068027138408182353680, −6.75592431218541628645429542668, −5.52046094859136981453357107237, −4.34994513526846678163280882622, −3.56020882484958113881535804224, −2.75403303708587302121914640120, −2.05129287630993450234477017003, −1.44109106318579000745856382817, 0,
1.44109106318579000745856382817, 2.05129287630993450234477017003, 2.75403303708587302121914640120, 3.56020882484958113881535804224, 4.34994513526846678163280882622, 5.52046094859136981453357107237, 6.75592431218541628645429542668, 7.30192869068027138408182353680, 7.72756384246078412459714876304
