| L(s) = 1 | − 3.15·2-s − 7.20·3-s + 1.96·4-s − 1.36·5-s + 22.7·6-s + 19.0·8-s + 24.9·9-s + 4.30·10-s + 64.7·11-s − 14.1·12-s + 19.2·13-s + 9.83·15-s − 75.8·16-s + 54.1·17-s − 78.8·18-s + 69.0·19-s − 2.67·20-s − 204.·22-s + 29.6·23-s − 137.·24-s − 123.·25-s − 60.9·26-s + 14.6·27-s + 13.1·29-s − 31.0·30-s − 185.·31-s + 87.0·32-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 1.38·3-s + 0.245·4-s − 0.121·5-s + 1.54·6-s + 0.842·8-s + 0.924·9-s + 0.136·10-s + 1.77·11-s − 0.340·12-s + 0.411·13-s + 0.169·15-s − 1.18·16-s + 0.772·17-s − 1.03·18-s + 0.833·19-s − 0.0299·20-s − 1.98·22-s + 0.268·23-s − 1.16·24-s − 0.985·25-s − 0.459·26-s + 0.104·27-s + 0.0840·29-s − 0.188·30-s − 1.07·31-s + 0.480·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 + 3.15T + 8T^{2} \) |
| 3 | \( 1 + 7.20T + 27T^{2} \) |
| 5 | \( 1 + 1.36T + 125T^{2} \) |
| 11 | \( 1 - 64.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 13.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 294.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 708.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 116.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 926.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 455.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 620.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 509.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 965.T + 9.12e5T^{2} \) |
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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
−8.555715082441763399216852964972, −7.47209152819928485928183921340, −6.94466007063413836671921907759, −6.04483577637070242789014143152, −5.35684535098985517169479894443, −4.34919698070238007852896829360, −3.52754931245059481400631823647, −1.60295937312262375707562144412, −1.01705447276979381840415802657, 0,
1.01705447276979381840415802657, 1.60295937312262375707562144412, 3.52754931245059481400631823647, 4.34919698070238007852896829360, 5.35684535098985517169479894443, 6.04483577637070242789014143152, 6.94466007063413836671921907759, 7.47209152819928485928183921340, 8.555715082441763399216852964972
