| L(s) = 1 | + 5.18·2-s + 18.9·4-s − 7·7-s + 56.5·8-s + 35.9·11-s − 45.2·13-s − 36.3·14-s + 142.·16-s + 113.·17-s + 61.5·19-s + 186.·22-s − 30.6·23-s − 234.·26-s − 132.·28-s − 214.·29-s + 164.·31-s + 284.·32-s + 586.·34-s + 410.·37-s + 319.·38-s + 309.·41-s + 29.9·43-s + 679.·44-s − 158.·46-s + 483.·47-s + 49·49-s − 855.·52-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.36·4-s − 0.377·7-s + 2.49·8-s + 0.985·11-s − 0.965·13-s − 0.693·14-s + 2.21·16-s + 1.61·17-s + 0.743·19-s + 1.80·22-s − 0.277·23-s − 1.77·26-s − 0.892·28-s − 1.37·29-s + 0.951·31-s + 1.57·32-s + 2.96·34-s + 1.82·37-s + 1.36·38-s + 1.17·41-s + 0.106·43-s + 2.32·44-s − 0.508·46-s + 1.50·47-s + 0.142·49-s − 2.28·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.041677397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.041677397\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.18T + 8T^{2} \) |
| 11 | \( 1 - 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 29.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 492.T + 9.12e5T^{2} \) |
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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
−9.315804689377933705655974523080, −7.69124759188804927838496979748, −7.35572397287503019672591970730, −6.21104799272102415133693837048, −5.78875778133228154217100808562, −4.84032720944667233693913709539, −4.02322816437731754010615951042, −3.25725567189747872944714527875, −2.40865164654261339916511378808, −1.10998961821647177357422447503,
1.10998961821647177357422447503, 2.40865164654261339916511378808, 3.25725567189747872944714527875, 4.02322816437731754010615951042, 4.84032720944667233693913709539, 5.78875778133228154217100808562, 6.21104799272102415133693837048, 7.35572397287503019672591970730, 7.69124759188804927838496979748, 9.315804689377933705655974523080
