| L(s) = 1 | − 2-s − 1.69·3-s + 4-s + 2.46·5-s + 1.69·6-s − 3.98·7-s − 8-s − 0.117·9-s − 2.46·10-s − 3.98·11-s − 1.69·12-s + 6.75·13-s + 3.98·14-s − 4.18·15-s + 16-s + 1.91·17-s + 0.117·18-s + 6.53·19-s + 2.46·20-s + 6.75·21-s + 3.98·22-s + 3.45·23-s + 1.69·24-s + 1.06·25-s − 6.75·26-s + 5.29·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.980·3-s + 0.5·4-s + 1.10·5-s + 0.693·6-s − 1.50·7-s − 0.353·8-s − 0.0391·9-s − 0.778·10-s − 1.20·11-s − 0.490·12-s + 1.87·13-s + 1.06·14-s − 1.07·15-s + 0.250·16-s + 0.464·17-s + 0.0276·18-s + 1.49·19-s + 0.550·20-s + 1.47·21-s + 0.850·22-s + 0.721·23-s + 0.346·24-s + 0.213·25-s − 1.32·26-s + 1.01·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7790281936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7790281936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 1.69T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 - 6.53T + 19T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + 0.603T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 3.80T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−8.648205612107498713547145827617, −7.65358919482056876304838713103, −6.79905472182148884574925396068, −6.18657254832375590768500046701, −5.63351012730783817083456448875, −5.24129551408978192270602277779, −3.45560735294492141293820628363, −3.01639840348029715284519309398, −1.68528825191041245536162681185, −0.59362569838371282262123255679,
0.59362569838371282262123255679, 1.68528825191041245536162681185, 3.01639840348029715284519309398, 3.45560735294492141293820628363, 5.24129551408978192270602277779, 5.63351012730783817083456448875, 6.18657254832375590768500046701, 6.79905472182148884574925396068, 7.65358919482056876304838713103, 8.648205612107498713547145827617
