| L(s) = 1 | + 3.24·3-s − 2.92·5-s + 3.52·7-s + 7.52·9-s + 5.72·11-s − 2.75·13-s − 9.47·15-s + 2.37·17-s + 5.03·19-s + 11.4·21-s − 5.86·23-s + 3.52·25-s + 14.6·27-s − 2.85·29-s + 10.2·31-s + 18.5·33-s − 10.3·35-s − 4.79·37-s − 8.94·39-s − 3.11·41-s + 11.7·43-s − 21.9·45-s + 4.95·47-s + 5.44·49-s + 7.70·51-s − 3.23·53-s − 16.7·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s − 1.30·5-s + 1.33·7-s + 2.50·9-s + 1.72·11-s − 0.764·13-s − 2.44·15-s + 0.575·17-s + 1.15·19-s + 2.49·21-s − 1.22·23-s + 0.705·25-s + 2.82·27-s − 0.531·29-s + 1.83·31-s + 3.23·33-s − 1.74·35-s − 0.788·37-s − 1.43·39-s − 0.487·41-s + 1.79·43-s − 3.27·45-s + 0.722·47-s + 0.777·49-s + 1.07·51-s − 0.443·53-s − 2.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.546462155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.546462155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 4.95T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 0.221T + 97T^{2} \) |
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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
−7.957231725475939831214180879914, −7.58402378512918131647101727043, −7.25622328890407163615683967962, −6.06538172899827102174609453905, −4.58858967658913590936410442830, −4.39217157180174417494730455653, −3.59258364151660216131667861869, −2.97071928814866050587871727239, −1.84795716217087670678704311643, −1.15856889757557798892551908032,
1.15856889757557798892551908032, 1.84795716217087670678704311643, 2.97071928814866050587871727239, 3.59258364151660216131667861869, 4.39217157180174417494730455653, 4.58858967658913590936410442830, 6.06538172899827102174609453905, 7.25622328890407163615683967962, 7.58402378512918131647101727043, 7.957231725475939831214180879914
