| L(s) = 1 | + 2.43·2-s − 0.331·3-s + 3.94·4-s − 0.0487·5-s − 0.807·6-s − 0.899·7-s + 4.72·8-s − 2.89·9-s − 0.118·10-s − 1.68·11-s − 1.30·12-s − 2.19·14-s + 0.0161·15-s + 3.64·16-s − 0.951·17-s − 7.04·18-s − 6.57·19-s − 0.192·20-s + 0.298·21-s − 4.11·22-s + 6.67·23-s − 1.56·24-s − 4.99·25-s + 1.95·27-s − 3.54·28-s + 8.28·29-s + 0.0393·30-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.191·3-s + 1.97·4-s − 0.0217·5-s − 0.329·6-s − 0.340·7-s + 1.67·8-s − 0.963·9-s − 0.0375·10-s − 0.509·11-s − 0.377·12-s − 0.586·14-s + 0.00417·15-s + 0.911·16-s − 0.230·17-s − 1.66·18-s − 1.50·19-s − 0.0429·20-s + 0.0650·21-s − 0.877·22-s + 1.39·23-s − 0.319·24-s − 0.999·25-s + 0.375·27-s − 0.669·28-s + 1.53·29-s + 0.00718·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.331T + 3T^{2} \) |
| 5 | \( 1 + 0.0487T + 5T^{2} \) |
| 7 | \( 1 + 0.899T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 17 | \( 1 + 0.951T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 37 | \( 1 + 6.36T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 + 7.18T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 + 8.60T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 1.14T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−7.56469584767783477716721563285, −6.73547638530974594398166715717, −6.19500056822891479911865761336, −5.63812559146964947851483131459, −4.80593338277312943453368365599, −4.34385400603725787363582913020, −3.20692094256236841998273160494, −2.85126898157711473860343004638, −1.83994245541378667233317661941, 0,
1.83994245541378667233317661941, 2.85126898157711473860343004638, 3.20692094256236841998273160494, 4.34385400603725787363582913020, 4.80593338277312943453368365599, 5.63812559146964947851483131459, 6.19500056822891479911865761336, 6.73547638530974594398166715717, 7.56469584767783477716721563285
