| L(s) = 1 | − 2-s + 1.69·3-s + 4-s − 3.98·5-s − 1.69·6-s + 1.28·7-s − 8-s − 0.118·9-s + 3.98·10-s − 11-s + 1.69·12-s + 4.42·13-s − 1.28·14-s − 6.76·15-s + 16-s − 6.41·17-s + 0.118·18-s + 1.08·19-s − 3.98·20-s + 2.18·21-s + 22-s + 1.35·23-s − 1.69·24-s + 10.8·25-s − 4.42·26-s − 5.29·27-s + 1.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.980·3-s + 0.5·4-s − 1.78·5-s − 0.692·6-s + 0.486·7-s − 0.353·8-s − 0.0395·9-s + 1.26·10-s − 0.301·11-s + 0.490·12-s + 1.22·13-s − 0.344·14-s − 1.74·15-s + 0.250·16-s − 1.55·17-s + 0.0279·18-s + 0.249·19-s − 0.891·20-s + 0.477·21-s + 0.213·22-s + 0.283·23-s − 0.346·24-s + 2.17·25-s − 0.867·26-s − 1.01·27-s + 0.243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 3.98T + 5T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 + 8.59T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 + 8.49T + 41T^{2} \) |
| 47 | \( 1 - 0.462T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 0.764T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−9.162286471292910803360940356547, −8.670720273889026711120282544674, −8.134180058297002720268398450200, −7.46850498457945898799186477049, −6.62079514887703910360358385706, −5.08601819736114591569017081114, −3.83660486465447810165843319001, −3.28462887016213615496001663395, −1.86382927924188150434996900298, 0,
1.86382927924188150434996900298, 3.28462887016213615496001663395, 3.83660486465447810165843319001, 5.08601819736114591569017081114, 6.62079514887703910360358385706, 7.46850498457945898799186477049, 8.134180058297002720268398450200, 8.670720273889026711120282544674, 9.162286471292910803360940356547
