| L(s) = 1 | − 0.488·3-s + 2.64·5-s − 2.76·9-s + 3.81·11-s − 2.51·13-s − 1.29·15-s − 5.06·17-s − 7.13·19-s + 23-s + 1.99·25-s + 2.81·27-s + 0.923·29-s + 5.59·31-s − 1.86·33-s + 5.39·37-s + 1.22·39-s + 1.47·41-s + 4.10·43-s − 7.30·45-s + 4.00·47-s + 2.47·51-s − 4.62·53-s + 10.0·55-s + 3.48·57-s − 2.40·59-s + 10.3·61-s − 6.64·65-s + ⋯ |
| L(s) = 1 | − 0.281·3-s + 1.18·5-s − 0.920·9-s + 1.14·11-s − 0.696·13-s − 0.333·15-s − 1.22·17-s − 1.63·19-s + 0.208·23-s + 0.398·25-s + 0.541·27-s + 0.171·29-s + 1.00·31-s − 0.324·33-s + 0.887·37-s + 0.196·39-s + 0.230·41-s + 0.626·43-s − 1.08·45-s + 0.584·47-s + 0.346·51-s − 0.635·53-s + 1.35·55-s + 0.461·57-s − 0.313·59-s + 1.33·61-s − 0.823·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.488T + 3T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 7.13T + 19T^{2} \) |
| 29 | \( 1 - 0.923T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 5.39T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 - 4.00T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 + 2.40T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 3.61T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 1.54T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.15601109105998799394333616230, −6.47668979112286891737457966906, −6.15785006114512159242423144072, −5.48067188552369973144964202063, −4.56291775089608626147686651622, −4.07569200168161210420374000388, −2.69696652502768392045908018008, −2.33433213625101394331547274488, −1.30714796775891183264418727371, 0,
1.30714796775891183264418727371, 2.33433213625101394331547274488, 2.69696652502768392045908018008, 4.07569200168161210420374000388, 4.56291775089608626147686651622, 5.48067188552369973144964202063, 6.15785006114512159242423144072, 6.47668979112286891737457966906, 7.15601109105998799394333616230
