| L(s) = 1 | − 2.10·2-s + 2.41·4-s + 3.44·5-s − 1.58·7-s − 0.875·8-s − 7.24·10-s + 1.37·11-s − 4.55·13-s + 3.32·14-s − 2.99·16-s − 4.04·17-s − 6.10·19-s + 8.33·20-s − 2.89·22-s + 6.90·25-s + 9.56·26-s − 3.82·28-s + 10.0·29-s + 1.13·31-s + 8.04·32-s + 8.50·34-s − 5.45·35-s + 8.26·37-s + 12.8·38-s − 3.02·40-s − 4.90·41-s + 2.87·43-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.20·4-s + 1.54·5-s − 0.597·7-s − 0.309·8-s − 2.29·10-s + 0.416·11-s − 1.26·13-s + 0.888·14-s − 0.748·16-s − 0.981·17-s − 1.40·19-s + 1.86·20-s − 0.618·22-s + 1.38·25-s + 1.87·26-s − 0.722·28-s + 1.87·29-s + 0.204·31-s + 1.42·32-s + 1.45·34-s − 0.922·35-s + 1.35·37-s + 2.08·38-s − 0.477·40-s − 0.765·41-s + 0.438·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 4.66T + 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 + 0.947T + 59T^{2} \) |
| 61 | \( 1 + 0.762T + 61T^{2} \) |
| 67 | \( 1 - 2.79T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 - 0.910T + 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + 3.66T + 89T^{2} \) |
| 97 | \( 1 - 1.70T + 97T^{2} \) |
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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
−8.248493932293037288536959020932, −7.16925666882668394859665970449, −6.51079067880238968972989388054, −6.25269382903903528169979199054, −5.01755967213260212055343938231, −4.31145524044280228524502292289, −2.64274424032957420873218738334, −2.29132169931267885233746632414, −1.27513712342546527237256027695, 0,
1.27513712342546527237256027695, 2.29132169931267885233746632414, 2.64274424032957420873218738334, 4.31145524044280228524502292289, 5.01755967213260212055343938231, 6.25269382903903528169979199054, 6.51079067880238968972989388054, 7.16925666882668394859665970449, 8.248493932293037288536959020932
