| L(s) = 1 | + 1.78·5-s + 0.671·7-s − 2.23·11-s + 0.0525·13-s − 5.52·17-s + 1.49·19-s + 7.13·23-s − 1.81·25-s − 2.08·29-s − 9.61·31-s + 1.19·35-s + 5.33·37-s + 11.8·41-s − 9.09·43-s − 5.21·47-s − 6.54·49-s − 4.27·53-s − 3.98·55-s + 0.163·59-s + 11.4·61-s + 0.0937·65-s − 10.6·67-s + 8.24·71-s − 12.8·73-s − 1.49·77-s − 7.86·79-s + 5.22·83-s + ⋯ |
| L(s) = 1 | + 0.798·5-s + 0.253·7-s − 0.672·11-s + 0.0145·13-s − 1.34·17-s + 0.343·19-s + 1.48·23-s − 0.362·25-s − 0.387·29-s − 1.72·31-s + 0.202·35-s + 0.876·37-s + 1.85·41-s − 1.38·43-s − 0.760·47-s − 0.935·49-s − 0.587·53-s − 0.536·55-s + 0.0213·59-s + 1.46·61-s + 0.0116·65-s − 1.30·67-s + 0.978·71-s − 1.50·73-s − 0.170·77-s − 0.885·79-s + 0.573·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.671T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 0.0525T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 23 | \( 1 - 7.13T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 - 5.33T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 4.27T + 53T^{2} \) |
| 59 | \( 1 - 0.163T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 8.24T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 - 5.22T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.40552691527630932080785601906, −6.68171310415027450514468376842, −6.00666484624013489937022419215, −5.27879697902607183527126707523, −4.78496144020206313830119482362, −3.86017951823340479084619227570, −2.89813439238644333505680131488, −2.19309690604485020238790904467, −1.38368904415655663943239163753, 0,
1.38368904415655663943239163753, 2.19309690604485020238790904467, 2.89813439238644333505680131488, 3.86017951823340479084619227570, 4.78496144020206313830119482362, 5.27879697902607183527126707523, 6.00666484624013489937022419215, 6.68171310415027450514468376842, 7.40552691527630932080785601906
