| L(s) = 1 | + 0.363·3-s − 5-s + 1.14·7-s − 2.86·9-s − 2.72·11-s − 4.64·13-s − 0.363·15-s − 0.858·17-s − 19-s + 0.414·21-s − 4.41·23-s + 25-s − 2.13·27-s + 9.42·29-s − 10.2·31-s − 0.990·33-s − 1.14·35-s + 6.77·37-s − 1.68·39-s − 7.55·41-s + 9.29·43-s + 2.86·45-s − 7.00·47-s − 5.69·49-s − 0.311·51-s − 8.64·53-s + 2.72·55-s + ⋯ |
| L(s) = 1 | + 0.209·3-s − 0.447·5-s + 0.431·7-s − 0.955·9-s − 0.822·11-s − 1.28·13-s − 0.0938·15-s − 0.208·17-s − 0.229·19-s + 0.0904·21-s − 0.920·23-s + 0.200·25-s − 0.410·27-s + 1.74·29-s − 1.84·31-s − 0.172·33-s − 0.192·35-s + 1.11·37-s − 0.270·39-s − 1.18·41-s + 1.41·43-s + 0.427·45-s − 1.02·47-s − 0.813·49-s − 0.0436·51-s − 1.18·53-s + 0.367·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.363T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 0.858T + 17T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 - 9.45T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.68T + 97T^{2} \) |
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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
−9.930978514218827801611978907719, −8.971385524946926485274035941654, −8.055160041239651651367071756563, −7.60657153408733379963511641028, −6.38735628882411981497300104999, −5.28914654442083821422453026470, −4.51705102928421661060724424478, −3.15759946591446452613300504056, −2.19209759456665165312882172637, 0,
2.19209759456665165312882172637, 3.15759946591446452613300504056, 4.51705102928421661060724424478, 5.28914654442083821422453026470, 6.38735628882411981497300104999, 7.60657153408733379963511641028, 8.055160041239651651367071756563, 8.971385524946926485274035941654, 9.930978514218827801611978907719
