| L(s) = 1 | − 1.90·2-s − 1.05·3-s + 2.64·4-s − 1.37·5-s + 2.01·6-s − 7-s − 3.13·8-s + 0.119·9-s + 2.62·10-s + 1.83·11-s − 2.79·12-s + 1.22·13-s + 1.90·14-s + 1.45·15-s + 3.33·16-s − 0.227·18-s − 3.63·20-s + 1.05·21-s − 3.50·22-s − 1.74·23-s + 3.31·24-s + 0.898·25-s − 2.33·26-s + 0.931·27-s − 2.64·28-s − 1.51·29-s − 2.78·30-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.05·3-s + 2.64·4-s − 1.37·5-s + 2.01·6-s − 7-s − 3.13·8-s + 0.119·9-s + 2.62·10-s + 1.83·11-s − 2.79·12-s + 1.22·13-s + 1.90·14-s + 1.45·15-s + 3.33·16-s − 0.227·18-s − 3.63·20-s + 1.05·21-s − 3.50·22-s − 1.74·23-s + 3.31·24-s + 0.898·25-s − 2.33·26-s + 0.931·27-s − 2.64·28-s − 1.51·29-s − 2.78·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1719408368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1719408368\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90T + T^{2} \) |
| 3 | \( 1 + 1.05T + T^{2} \) |
| 5 | \( 1 + 1.37T + T^{2} \) |
| 11 | \( 1 - 1.83T + T^{2} \) |
| 13 | \( 1 - 1.22T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.74T + T^{2} \) |
| 29 | \( 1 + 1.51T + T^{2} \) |
| 31 | \( 1 + 1.95T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.302T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.101T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.880T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.052481204898696071665755219196, −8.550002454897730909331591763652, −7.59698171685314966101556788548, −7.01143616724707544216163573547, −6.21081179507388076801866245860, −5.88227547515651495011093578773, −3.89492464597998503585595700563, −3.50543964103880562386185360452, −1.75847692962761879104321981230, −0.54089345236999443251677177725,
0.54089345236999443251677177725, 1.75847692962761879104321981230, 3.50543964103880562386185360452, 3.89492464597998503585595700563, 5.88227547515651495011093578773, 6.21081179507388076801866245860, 7.01143616724707544216163573547, 7.59698171685314966101556788548, 8.550002454897730909331591763652, 9.052481204898696071665755219196
