| L(s) = 1 | + 1.57·2-s + 1.09·3-s + 1.49·4-s − 0.165·5-s + 1.72·6-s + 1.57·7-s + 0.774·8-s + 0.196·9-s − 0.260·10-s − 0.165·11-s + 1.63·12-s − 0.803·13-s + 2.49·14-s − 0.180·15-s − 0.267·16-s + 0.310·18-s − 1.35·19-s − 0.246·20-s + 1.72·21-s − 0.260·22-s + 0.847·24-s − 0.972·25-s − 1.26·26-s − 0.878·27-s + 2.35·28-s − 0.285·30-s − 1.19·32-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.09·3-s + 1.49·4-s − 0.165·5-s + 1.72·6-s + 1.57·7-s + 0.774·8-s + 0.196·9-s − 0.260·10-s − 0.165·11-s + 1.63·12-s − 0.803·13-s + 2.49·14-s − 0.180·15-s − 0.267·16-s + 0.310·18-s − 1.35·19-s − 0.246·20-s + 1.72·21-s − 0.260·22-s + 0.847·24-s − 0.972·25-s − 1.26·26-s − 0.878·27-s + 2.35·28-s − 0.285·30-s − 1.19·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.175823077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.175823077\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| good | 2 | \( 1 - 1.57T + T^{2} \) |
| 3 | \( 1 - 1.09T + T^{2} \) |
| 5 | \( 1 + 0.165T + T^{2} \) |
| 7 | \( 1 - 1.57T + T^{2} \) |
| 11 | \( 1 + 0.165T + T^{2} \) |
| 13 | \( 1 + 0.803T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.35T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.803T + T^{2} \) |
| 43 | \( 1 - 0.490T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.89T + T^{2} \) |
| 89 | \( 1 - 1.09T + T^{2} \) |
| 97 | \( 1 - 1.09T + T^{2} \) |
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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
−8.680900267758392884787555944732, −7.977157472879491414852763918779, −7.52085729372078656687208450099, −6.46405027152919036119856554027, −5.57990621226135912391945647971, −4.80866268746380365058732878744, −4.25585111757946102269027943245, −3.47422603321756887312079677806, −2.38783159635550580122095103327, −2.00529257537597998204374337251,
2.00529257537597998204374337251, 2.38783159635550580122095103327, 3.47422603321756887312079677806, 4.25585111757946102269027943245, 4.80866268746380365058732878744, 5.57990621226135912391945647971, 6.46405027152919036119856554027, 7.52085729372078656687208450099, 7.977157472879491414852763918779, 8.680900267758392884787555944732
