| L(s) = 1 | + 2-s + 3-s + 4-s + 1.34·5-s + 6-s + 7-s + 8-s + 9-s + 1.34·10-s + 4.09·11-s + 12-s − 3.44·13-s + 14-s + 1.34·15-s + 16-s + 2.65·17-s + 18-s + 2.09·19-s + 1.34·20-s + 21-s + 4.09·22-s − 8.09·23-s + 24-s − 3.19·25-s − 3.44·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.601·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.424·10-s + 1.23·11-s + 0.288·12-s − 0.954·13-s + 0.267·14-s + 0.347·15-s + 0.250·16-s + 0.644·17-s + 0.235·18-s + 0.481·19-s + 0.300·20-s + 0.218·21-s + 0.873·22-s − 1.68·23-s + 0.204·24-s − 0.638·25-s − 0.674·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.687476103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.687476103\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 + 8.09T + 23T^{2} \) |
| 31 | \( 1 + 3.34T + 31T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 8.12T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 0.752T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 8.16T + 89T^{2} \) |
| 97 | \( 1 - 8.94T + 97T^{2} \) |
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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
−9.640031262094455284934456938995, −9.116731630154823689914095857978, −7.86234208456716263717901181062, −7.34726062446322107183241182200, −6.21085562553299244012248502863, −5.56073082875701947241359741470, −4.41534629123774776951750642638, −3.68559072449978358320384621959, −2.46043114846623691769507837514, −1.55032299970274012195778986266,
1.55032299970274012195778986266, 2.46043114846623691769507837514, 3.68559072449978358320384621959, 4.41534629123774776951750642638, 5.56073082875701947241359741470, 6.21085562553299244012248502863, 7.34726062446322107183241182200, 7.86234208456716263717901181062, 9.116731630154823689914095857978, 9.640031262094455284934456938995
