| L(s) = 1 | + 2.19·2-s + 2.48·3-s + 2.81·4-s + 5.46·6-s − 3.05·7-s + 1.78·8-s + 3.19·9-s − 5.36·11-s + 7.00·12-s + 4.59·13-s − 6.70·14-s − 1.70·16-s − 17-s + 7.01·18-s + 4.57·19-s − 7.60·21-s − 11.7·22-s − 1.24·23-s + 4.45·24-s + 10.0·26-s + 0.483·27-s − 8.60·28-s − 5.93·29-s + 9.84·31-s − 7.31·32-s − 13.3·33-s − 2.19·34-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.43·3-s + 1.40·4-s + 2.22·6-s − 1.15·7-s + 0.632·8-s + 1.06·9-s − 1.61·11-s + 2.02·12-s + 1.27·13-s − 1.79·14-s − 0.425·16-s − 0.242·17-s + 1.65·18-s + 1.04·19-s − 1.66·21-s − 2.50·22-s − 0.260·23-s + 0.909·24-s + 1.97·26-s + 0.0931·27-s − 1.62·28-s − 1.10·29-s + 1.76·31-s − 1.29·32-s − 2.32·33-s − 0.376·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.018415496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.018415496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 - 9.84T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 0.404T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 4.97T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.27T + 97T^{2} \) |
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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
−11.40872411642811798759546356689, −10.28426975541938442285083865593, −9.336705745263963062400107322153, −8.352679768472228316664666784343, −7.38117201711609608836389274984, −6.25624254535781492607334806829, −5.31735459504638279298642685826, −3.92917325756505295924134791008, −3.19616275514473665401667350554, −2.46975049614380243423524933164,
2.46975049614380243423524933164, 3.19616275514473665401667350554, 3.92917325756505295924134791008, 5.31735459504638279298642685826, 6.25624254535781492607334806829, 7.38117201711609608836389274984, 8.352679768472228316664666784343, 9.336705745263963062400107322153, 10.28426975541938442285083865593, 11.40872411642811798759546356689
