| L(s) = 1 | − 1.34·3-s + 3.90·5-s + 0.487·7-s − 1.19·9-s − 4.04·11-s + 3.53·13-s − 5.24·15-s − 6.37·17-s − 0.975·19-s − 0.655·21-s − 0.750·23-s + 10.2·25-s + 5.63·27-s − 4.08·29-s + 7.86·31-s + 5.44·33-s + 1.90·35-s − 4.22·37-s − 4.75·39-s − 2.07·41-s − 10.7·43-s − 4.66·45-s − 9.31·47-s − 6.76·49-s + 8.56·51-s + 11.1·53-s − 15.8·55-s + ⋯ |
| L(s) = 1 | − 0.776·3-s + 1.74·5-s + 0.184·7-s − 0.397·9-s − 1.22·11-s + 0.981·13-s − 1.35·15-s − 1.54·17-s − 0.223·19-s − 0.143·21-s − 0.156·23-s + 2.05·25-s + 1.08·27-s − 0.757·29-s + 1.41·31-s + 0.947·33-s + 0.322·35-s − 0.694·37-s − 0.761·39-s − 0.323·41-s − 1.63·43-s − 0.694·45-s − 1.35·47-s − 0.965·49-s + 1.19·51-s + 1.52·53-s − 2.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 - 0.487T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 + 0.975T + 19T^{2} \) |
| 23 | \( 1 + 0.750T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 9.31T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 + 7.53T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.75T + 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 + 9.63T + 97T^{2} \) |
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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
−8.293681672390597665803250741591, −6.95902862183388946670158939829, −6.32859393734845081222928972925, −5.88625135694040973037667429887, −5.15446269506707991558664823736, −4.63409495869031594267188865522, −3.15273637781708652618152298017, −2.31402949087458923211614839918, −1.50539721523385660141825486145, 0,
1.50539721523385660141825486145, 2.31402949087458923211614839918, 3.15273637781708652618152298017, 4.63409495869031594267188865522, 5.15446269506707991558664823736, 5.88625135694040973037667429887, 6.32859393734845081222928972925, 6.95902862183388946670158939829, 8.293681672390597665803250741591
