| L(s) = 1 | + 3-s − 5-s − 2.48·7-s + 9-s − 0.387·11-s + 2.48·13-s − 15-s − 0.418·17-s − 4·19-s − 2.48·21-s + 5.11·23-s + 25-s + 27-s − 4.24·29-s − 31-s − 0.387·33-s + 2.48·35-s + 3.83·37-s + 2.48·39-s + 4.31·41-s − 6·43-s − 45-s − 10.8·47-s − 0.843·49-s − 0.418·51-s − 8.85·53-s + 0.387·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.937·7-s + 0.333·9-s − 0.116·11-s + 0.688·13-s − 0.258·15-s − 0.101·17-s − 0.917·19-s − 0.541·21-s + 1.06·23-s + 0.200·25-s + 0.192·27-s − 0.789·29-s − 0.179·31-s − 0.0675·33-s + 0.419·35-s + 0.629·37-s + 0.397·39-s + 0.673·41-s − 0.914·43-s − 0.149·45-s − 1.58·47-s − 0.120·49-s − 0.0585·51-s − 1.21·53-s + 0.0523·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 + 0.387T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 + 0.418T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + 3.86T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 + 4.60T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 5.89T + 83T^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.237346912691095282643228644248, −7.45438476595171983408116883124, −6.67860909478040035067302040486, −6.12048277553750258296160034511, −5.02457931357527338414866918336, −4.12450733998618051804368024320, −3.39947419362783258276354749745, −2.71639421188040218634791006690, −1.48443147360890699640388357368, 0,
1.48443147360890699640388357368, 2.71639421188040218634791006690, 3.39947419362783258276354749745, 4.12450733998618051804368024320, 5.02457931357527338414866918336, 6.12048277553750258296160034511, 6.67860909478040035067302040486, 7.45438476595171983408116883124, 8.237346912691095282643228644248
