| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 11-s + 12-s − 7.12·13-s − 15-s + 16-s − 17-s + 18-s + 3.12·19-s − 20-s − 22-s − 0.876·23-s + 24-s + 25-s − 7.12·26-s + 27-s − 0.876·29-s − 30-s + 1.12·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.97·13-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.716·19-s − 0.223·20-s − 0.213·22-s − 0.182·23-s + 0.204·24-s + 0.200·25-s − 1.39·26-s + 0.192·27-s − 0.162·29-s − 0.182·30-s + 0.201·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 0.876T + 23T^{2} \) |
| 29 | \( 1 + 0.876T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 0.246T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−7.52758535829043867382635461418, −7.26897996237402212074464893435, −6.43963903677969449862242740243, −5.32907371979394353898393743050, −4.87569358316474075803455692966, −4.10662162864964658369848263162, −3.18949179672656999399838934485, −2.61070292968244389297295922065, −1.66442771112067544496094532083, 0,
1.66442771112067544496094532083, 2.61070292968244389297295922065, 3.18949179672656999399838934485, 4.10662162864964658369848263162, 4.87569358316474075803455692966, 5.32907371979394353898393743050, 6.43963903677969449862242740243, 7.26897996237402212074464893435, 7.52758535829043867382635461418
