| L(s) = 1 | + 2.39·2-s + 3.71·4-s + 5-s + 1.95·7-s + 4.09·8-s + 2.39·10-s + 4.29·11-s + 1.05·13-s + 4.67·14-s + 2.36·16-s − 3.65·17-s − 3.07·19-s + 3.71·20-s + 10.2·22-s + 1.46·23-s + 25-s + 2.51·26-s + 7.25·28-s + 8.44·29-s + 4.24·31-s − 2.54·32-s − 8.73·34-s + 1.95·35-s − 4.61·37-s − 7.35·38-s + 4.09·40-s − 0.785·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.85·4-s + 0.447·5-s + 0.738·7-s + 1.44·8-s + 0.755·10-s + 1.29·11-s + 0.292·13-s + 1.24·14-s + 0.590·16-s − 0.886·17-s − 0.706·19-s + 0.830·20-s + 2.18·22-s + 0.305·23-s + 0.200·25-s + 0.493·26-s + 1.37·28-s + 1.56·29-s + 0.761·31-s − 0.449·32-s − 1.49·34-s + 0.330·35-s − 0.758·37-s − 1.19·38-s + 0.647·40-s − 0.122·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.838509254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.838509254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 + 0.785T + 41T^{2} \) |
| 43 | \( 1 + 3.60T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 - 5.36T + 59T^{2} \) |
| 61 | \( 1 + 5.59T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 2.47T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
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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
−8.631743654010530377106866067796, −7.31384729028771310471710090547, −6.61800666850569834725955469567, −6.19118561416418827914855255884, −5.35414523167311294253984538633, −4.48221434061278478404093098781, −4.19968384219659426395014150928, −3.11695519039676024657672729734, −2.24257428578013519286376568567, −1.33689810125117329232002439819,
1.33689810125117329232002439819, 2.24257428578013519286376568567, 3.11695519039676024657672729734, 4.19968384219659426395014150928, 4.48221434061278478404093098781, 5.35414523167311294253984538633, 6.19118561416418827914855255884, 6.61800666850569834725955469567, 7.31384729028771310471710090547, 8.631743654010530377106866067796
