L(s) = 1 | − 2-s − 0.418·3-s + 4-s + 5-s + 0.418·6-s − 7-s − 8-s − 2.82·9-s − 10-s − 0.418·12-s + 5.30·13-s + 14-s − 0.418·15-s + 16-s − 1.47·17-s + 2.82·18-s − 0.952·19-s + 20-s + 0.418·21-s + 2.37·23-s + 0.418·24-s + 25-s − 5.30·26-s + 2.43·27-s − 28-s − 5.97·29-s + 0.418·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.241·3-s + 0.5·4-s + 0.447·5-s + 0.171·6-s − 0.377·7-s − 0.353·8-s − 0.941·9-s − 0.316·10-s − 0.120·12-s + 1.47·13-s + 0.267·14-s − 0.108·15-s + 0.250·16-s − 0.358·17-s + 0.665·18-s − 0.218·19-s + 0.223·20-s + 0.0914·21-s + 0.495·23-s + 0.0855·24-s + 0.200·25-s − 1.03·26-s + 0.469·27-s − 0.188·28-s − 1.11·29-s + 0.0764·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 0.418T + 3T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 0.952T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 - 0.0224T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 + 1.50T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 0.768T + 59T^{2} \) |
| 61 | \( 1 + 8.06T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 6.13T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−7.53206709471357863644512527863, −6.58315514364810615425262181109, −6.20905748092533099970048839580, −5.62994009779115965708311334764, −4.76076565541227259024772181535, −3.63926556877605486613028881474, −3.02697828856266508585647460515, −2.07357966365610978109884131407, −1.14365062572257572899502140903, 0,
1.14365062572257572899502140903, 2.07357966365610978109884131407, 3.02697828856266508585647460515, 3.63926556877605486613028881474, 4.76076565541227259024772181535, 5.62994009779115965708311334764, 6.20905748092533099970048839580, 6.58315514364810615425262181109, 7.53206709471357863644512527863