| L(s) = 1 | + 1.73·2-s − 3-s + 0.999·4-s − 5-s − 1.73·6-s − 1.73·7-s − 1.73·8-s − 2·9-s − 1.73·10-s − 0.999·12-s + 3.46·13-s − 2.99·14-s + 15-s − 5·16-s − 6.92·17-s − 3.46·18-s − 3.46·19-s − 0.999·20-s + 1.73·21-s + 1.73·24-s + 25-s + 5.99·26-s + 5·27-s − 1.73·28-s + 1.73·30-s − 8·31-s − 5.19·32-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.447·5-s − 0.707·6-s − 0.654·7-s − 0.612·8-s − 0.666·9-s − 0.547·10-s − 0.288·12-s + 0.960·13-s − 0.801·14-s + 0.258·15-s − 1.25·16-s − 1.68·17-s − 0.816·18-s − 0.794·19-s − 0.223·20-s + 0.377·21-s + 0.353·24-s + 0.200·25-s + 1.17·26-s + 0.962·27-s − 0.327·28-s + 0.316·30-s − 1.43·31-s − 0.918·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.66T + 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.81667643402762384082211745565, −9.097081782258434538096454156701, −8.696146579504793606544450782183, −7.12738424913006641520490750420, −6.21582976908259805341697702925, −5.69161695820892943497683929412, −4.48129188551813999289632652079, −3.75162009716497306769710738140, −2.57845760710912345016843008396, 0,
2.57845760710912345016843008396, 3.75162009716497306769710738140, 4.48129188551813999289632652079, 5.69161695820892943497683929412, 6.21582976908259805341697702925, 7.12738424913006641520490750420, 8.696146579504793606544450782183, 9.097081782258434538096454156701, 10.81667643402762384082211745565
