| L(s) = 1 | + 1.76·5-s − 3.18·7-s + 0.181·11-s − 3.18·13-s + 7.84·17-s + 19-s − 3.28·23-s − 1.89·25-s + 9.48·29-s − 2.18·31-s − 5.60·35-s − 6.42·37-s + 1.76·41-s − 9.16·43-s − 9·47-s + 3.12·49-s − 13.6·53-s + 0.320·55-s + 8.78·59-s − 0.703·61-s − 5.60·65-s + 6.37·67-s + 3.22·71-s − 6.88·73-s − 0.578·77-s + 0.986·79-s + 10.2·83-s + ⋯ |
| L(s) = 1 | + 0.787·5-s − 1.20·7-s + 0.0548·11-s − 0.882·13-s + 1.90·17-s + 0.229·19-s − 0.684·23-s − 0.379·25-s + 1.76·29-s − 0.391·31-s − 0.947·35-s − 1.05·37-s + 0.275·41-s − 1.39·43-s − 1.31·47-s + 0.446·49-s − 1.87·53-s + 0.0432·55-s + 1.14·59-s − 0.0900·61-s − 0.694·65-s + 0.779·67-s + 0.382·71-s − 0.805·73-s − 0.0659·77-s + 0.110·79-s + 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 0.181T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 - 7.84T + 17T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 2.18T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 + 0.703T + 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 3.22T + 71T^{2} \) |
| 73 | \( 1 + 6.88T + 73T^{2} \) |
| 79 | \( 1 - 0.986T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 7.58T + 89T^{2} \) |
| 97 | \( 1 - 0.784T + 97T^{2} \) |
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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
−7.49950381733190731818484506948, −6.52114580519696065430561490931, −6.28539214187917073244216346804, −5.34948365675017299770808761023, −4.89650723493076761782740310491, −3.60854478233548054226515639024, −3.18099908340865732782881926471, −2.27465465774466991957711951952, −1.28635348977715764021152980952, 0,
1.28635348977715764021152980952, 2.27465465774466991957711951952, 3.18099908340865732782881926471, 3.60854478233548054226515639024, 4.89650723493076761782740310491, 5.34948365675017299770808761023, 6.28539214187917073244216346804, 6.52114580519696065430561490931, 7.49950381733190731818484506948
