| L(s) = 1 | + 2.57·2-s + 3-s + 4.61·4-s − 3.47·5-s + 2.57·6-s − 7-s + 6.72·8-s + 9-s − 8.92·10-s + 4.38·11-s + 4.61·12-s − 0.503·13-s − 2.57·14-s − 3.47·15-s + 8.06·16-s + 1.70·17-s + 2.57·18-s + 2.45·19-s − 16.0·20-s − 21-s + 11.2·22-s + 5.47·23-s + 6.72·24-s + 7.05·25-s − 1.29·26-s + 27-s − 4.61·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.30·4-s − 1.55·5-s + 1.05·6-s − 0.377·7-s + 2.37·8-s + 0.333·9-s − 2.82·10-s + 1.32·11-s + 1.33·12-s − 0.139·13-s − 0.687·14-s − 0.896·15-s + 2.01·16-s + 0.412·17-s + 0.606·18-s + 0.562·19-s − 3.58·20-s − 0.218·21-s + 2.40·22-s + 1.14·23-s + 1.37·24-s + 1.41·25-s − 0.253·26-s + 0.192·27-s − 0.872·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.292806220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.292806220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 5 | \( 1 + 3.47T + 5T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 + 0.503T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 - 8.91T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 7.20T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 + 0.651T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 + 9.02T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 + 0.689T + 89T^{2} \) |
| 97 | \( 1 + 9.03T + 97T^{2} \) |
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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
−8.181748494807326898326672109030, −7.39710038124801830373102793515, −6.93665528583038283118172558465, −6.23710719343009084812080934691, −5.18260747283603374858567966701, −4.45562370218974740135813834738, −3.74870184225914732054579417534, −3.41941412869039525213096142402, −2.56443253893207260531478123487, −1.14669189671325811056900422031,
1.14669189671325811056900422031, 2.56443253893207260531478123487, 3.41941412869039525213096142402, 3.74870184225914732054579417534, 4.45562370218974740135813834738, 5.18260747283603374858567966701, 6.23710719343009084812080934691, 6.93665528583038283118172558465, 7.39710038124801830373102793515, 8.181748494807326898326672109030
