| L(s) = 1 | − 2-s + 1.91·3-s + 4-s − 5-s − 1.91·6-s − 7-s − 8-s + 0.683·9-s + 10-s + 1.91·12-s − 0.869·13-s + 14-s − 1.91·15-s + 16-s − 4.15·17-s − 0.683·18-s + 7.56·19-s − 20-s − 1.91·21-s − 2·23-s − 1.91·24-s + 25-s + 0.869·26-s − 4.44·27-s − 28-s − 1.19·29-s + 1.91·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.10·3-s + 0.5·4-s − 0.447·5-s − 0.783·6-s − 0.377·7-s − 0.353·8-s + 0.227·9-s + 0.316·10-s + 0.554·12-s − 0.241·13-s + 0.267·14-s − 0.495·15-s + 0.250·16-s − 1.00·17-s − 0.160·18-s + 1.73·19-s − 0.223·20-s − 0.418·21-s − 0.417·23-s − 0.391·24-s + 0.200·25-s + 0.170·26-s − 0.855·27-s − 0.188·28-s − 0.222·29-s + 0.350·30-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 | \( 1 - 1.91T + 3T^{2} \) |
| 13 | \( 1 + 0.869T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 - 7.56T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 + 9.86T + 41T^{2} \) |
| 43 | \( 1 + 6.62T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.57T + 53T^{2} \) |
| 59 | \( 1 + 7.43T + 59T^{2} \) |
| 61 | \( 1 + 7.73T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 - 17.0T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.64827139422441836107861321580, −7.01332139697570283957541823408, −6.31467788723494188413836237421, −5.40874795183790324590013976173, −4.47141800036798768317385385863, −3.59686935022065176530043590751, −2.94977437651244492772893670245, −2.35008290452733810692599701387, −1.25639945374334753573577170555, 0,
1.25639945374334753573577170555, 2.35008290452733810692599701387, 2.94977437651244492772893670245, 3.59686935022065176530043590751, 4.47141800036798768317385385863, 5.40874795183790324590013976173, 6.31467788723494188413836237421, 7.01332139697570283957541823408, 7.64827139422441836107861321580
