| L(s) = 1 | − 3-s + 5-s + 4.42·7-s + 9-s − 11-s − 0.622·13-s − 15-s − 5.18·17-s − 7.05·19-s − 4.42·21-s − 8.85·23-s + 25-s − 27-s − 7.80·29-s − 2.75·31-s + 33-s + 4.42·35-s − 2·37-s + 0.622·39-s − 0.193·41-s − 5.67·43-s + 45-s + 2.75·47-s + 12.6·49-s + 5.18·51-s − 10.8·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.67·7-s + 0.333·9-s − 0.301·11-s − 0.172·13-s − 0.258·15-s − 1.25·17-s − 1.61·19-s − 0.966·21-s − 1.84·23-s + 0.200·25-s − 0.192·27-s − 1.44·29-s − 0.494·31-s + 0.174·33-s + 0.748·35-s − 0.328·37-s + 0.0996·39-s − 0.0302·41-s − 0.865·43-s + 0.149·45-s + 0.401·47-s + 1.80·49-s + 0.725·51-s − 1.49·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 0.193T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 - 4.23T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 + 0.133T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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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
−8.391663282320161713401167395065, −7.83046392536900667641767918779, −6.89460668168336957054908332059, −6.07917125953918005816771091669, −5.34478951113909419184355010781, −4.58637969334432607644709467657, −3.95952266915956602547684779065, −2.13306533034983871672222013848, −1.82290899663276006634985141032, 0,
1.82290899663276006634985141032, 2.13306533034983871672222013848, 3.95952266915956602547684779065, 4.58637969334432607644709467657, 5.34478951113909419184355010781, 6.07917125953918005816771091669, 6.89460668168336957054908332059, 7.83046392536900667641767918779, 8.391663282320161713401167395065
