| L(s) = 1 | − 0.740·2-s − 3.20·3-s − 1.45·4-s − 5-s + 2.37·6-s − 1.10·7-s + 2.55·8-s + 7.29·9-s + 0.740·10-s − 6.39·11-s + 4.65·12-s − 1.24·13-s + 0.816·14-s + 3.20·15-s + 1.01·16-s + 0.740·17-s − 5.40·18-s − 5.20·19-s + 1.45·20-s + 3.54·21-s + 4.73·22-s − 3.43·23-s − 8.19·24-s + 25-s + 0.920·26-s − 13.7·27-s + 1.60·28-s + ⋯ |
| L(s) = 1 | − 0.523·2-s − 1.85·3-s − 0.725·4-s − 0.447·5-s + 0.969·6-s − 0.417·7-s + 0.903·8-s + 2.43·9-s + 0.234·10-s − 1.92·11-s + 1.34·12-s − 0.344·13-s + 0.218·14-s + 0.828·15-s + 0.253·16-s + 0.179·17-s − 1.27·18-s − 1.19·19-s + 0.324·20-s + 0.772·21-s + 1.00·22-s − 0.716·23-s − 1.67·24-s + 0.200·25-s + 0.180·26-s − 2.65·27-s + 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.740T + 2T^{2} \) |
| 3 | \( 1 + 3.20T + 3T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 - 0.740T + 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 + 0.989T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 + 7.02T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 1.17T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.978413415619279855000693467541, −7.34935597537871855798728549141, −6.51217835102824784125204529482, −5.76514147104479460287049666699, −4.96616276627459320291590483925, −4.65884195793967107756301421095, −3.65576672640191465974392188092, −2.18116577683136409356419780355, −0.69306317151370819530566048954, 0,
0.69306317151370819530566048954, 2.18116577683136409356419780355, 3.65576672640191465974392188092, 4.65884195793967107756301421095, 4.96616276627459320291590483925, 5.76514147104479460287049666699, 6.51217835102824784125204529482, 7.34935597537871855798728549141, 7.978413415619279855000693467541
