| L(s) = 1 | + 5-s − 7-s + 1.61·11-s + 13-s − 5.85·17-s − 2.61·19-s + 2.23·23-s + 25-s − 5.85·29-s + 1.47·31-s − 35-s + 37-s + 8.70·41-s − 3.85·43-s + 8.70·47-s − 6·49-s − 1.52·53-s + 1.61·55-s − 9.79·59-s − 5.76·61-s + 65-s − 7.09·67-s − 0.326·71-s + 3.32·73-s − 1.61·77-s − 15.3·79-s − 11.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.487·11-s + 0.277·13-s − 1.41·17-s − 0.600·19-s + 0.466·23-s + 0.200·25-s − 1.08·29-s + 0.264·31-s − 0.169·35-s + 0.164·37-s + 1.35·41-s − 0.587·43-s + 1.27·47-s − 0.857·49-s − 0.209·53-s + 0.218·55-s − 1.27·59-s − 0.737·61-s + 0.124·65-s − 0.866·67-s − 0.0387·71-s + 0.389·73-s − 0.184·77-s − 1.72·79-s − 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 + 3.85T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 + 0.326T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 3.29T + 97T^{2} \) |
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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
−7.54788168966767605439243214037, −6.84841127448562744907726267903, −6.23558091986290589522349850621, −5.69868048231767154775982940077, −4.63036991325954210208499718029, −4.11305221907826381310759494866, −3.11437451706073462538376281723, −2.28366401027228059680996732698, −1.38190048355444808798250315201, 0,
1.38190048355444808798250315201, 2.28366401027228059680996732698, 3.11437451706073462538376281723, 4.11305221907826381310759494866, 4.63036991325954210208499718029, 5.69868048231767154775982940077, 6.23558091986290589522349850621, 6.84841127448562744907726267903, 7.54788168966767605439243214037
