| L(s) = 1 | − 1.75·3-s + 1.89·5-s − 3.72·7-s + 0.0728·9-s + 5.05·11-s + 3.05·13-s − 3.32·15-s + 0.674·17-s − 1.01·19-s + 6.53·21-s − 4.16·23-s − 1.40·25-s + 5.13·27-s − 6.08·29-s + 1.11·31-s − 8.86·33-s − 7.07·35-s − 8.11·37-s − 5.35·39-s + 9.59·41-s − 10.0·43-s + 0.138·45-s − 2.58·47-s + 6.91·49-s − 1.18·51-s + 8.96·53-s + 9.59·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s + 0.848·5-s − 1.40·7-s + 0.0242·9-s + 1.52·11-s + 0.847·13-s − 0.858·15-s + 0.163·17-s − 0.233·19-s + 1.42·21-s − 0.868·23-s − 0.280·25-s + 0.987·27-s − 1.12·29-s + 0.200·31-s − 1.54·33-s − 1.19·35-s − 1.33·37-s − 0.857·39-s + 1.49·41-s − 1.53·43-s + 0.0205·45-s − 0.377·47-s + 0.987·49-s − 0.165·51-s + 1.23·53-s + 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 0.674T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 8.96T + 53T^{2} \) |
| 59 | \( 1 + 4.15T + 59T^{2} \) |
| 61 | \( 1 - 7.74T + 61T^{2} \) |
| 67 | \( 1 + 4.80T + 67T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 + 0.804T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 0.249T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 6.87T + 97T^{2} \) |
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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
−7.50509681244885420685616881079, −6.59075906475965765304932137198, −6.22101481822865624731504746431, −5.91991078660050851080636183792, −5.05975750403468523235774058923, −3.87887900129805060993849695462, −3.46250695594827181343675619615, −2.19199053732584294834489196599, −1.19294278060123591951712247758, 0,
1.19294278060123591951712247758, 2.19199053732584294834489196599, 3.46250695594827181343675619615, 3.87887900129805060993849695462, 5.05975750403468523235774058923, 5.91991078660050851080636183792, 6.22101481822865624731504746431, 6.59075906475965765304932137198, 7.50509681244885420685616881079
