| L(s) = 1 | + 1.54·2-s + 0.381·4-s + 1.54·5-s − 0.236·7-s − 2.49·8-s + 2.38·10-s + 4.23·13-s − 0.364·14-s − 4.61·16-s + 5.94·17-s + 4.61·19-s + 0.589·20-s + 7.49·23-s − 2.61·25-s + 6.53·26-s − 0.0901·28-s − 1.90·29-s − 2.38·31-s − 2.13·32-s + 9.18·34-s − 0.364·35-s + 6.23·37-s + 7.12·38-s − 3.85·40-s − 5.58·41-s + 10.7·43-s + 11.5·46-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.190·4-s + 0.690·5-s − 0.0892·7-s − 0.882·8-s + 0.753·10-s + 1.17·13-s − 0.0973·14-s − 1.15·16-s + 1.44·17-s + 1.05·19-s + 0.131·20-s + 1.56·23-s − 0.523·25-s + 1.28·26-s − 0.0170·28-s − 0.354·29-s − 0.427·31-s − 0.377·32-s + 1.57·34-s − 0.0615·35-s + 1.02·37-s + 1.15·38-s − 0.609·40-s − 0.872·41-s + 1.63·43-s + 1.70·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.028009133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.028009133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 - 6.23T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.953T + 47T^{2} \) |
| 53 | \( 1 + 9.03T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 - 3.85T + 67T^{2} \) |
| 71 | \( 1 + 7.71T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 4.40T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.656813320704953835499239449511, −9.298199611234629912714140487841, −8.203329384540423732039058157585, −7.17473845795276278684006668830, −6.02834173101124052852634614832, −5.65553740590037072979261575016, −4.74473700953626583185500342180, −3.58543709616698756572202238016, −2.94173549873966310970190057883, −1.28989780591299017530238660629,
1.28989780591299017530238660629, 2.94173549873966310970190057883, 3.58543709616698756572202238016, 4.74473700953626583185500342180, 5.65553740590037072979261575016, 6.02834173101124052852634614832, 7.17473845795276278684006668830, 8.203329384540423732039058157585, 9.298199611234629912714140487841, 9.656813320704953835499239449511
