| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 5-s − 7-s + 2.23·8-s − 1.61·10-s + 1.23·11-s − 6.47·13-s + 1.61·14-s − 4.85·16-s − 0.236·17-s + 1.47·19-s + 0.618·20-s − 2.00·22-s − 1.76·23-s + 25-s + 10.4·26-s − 0.618·28-s + 5.70·29-s − 9·31-s + 3.38·32-s + 0.381·34-s − 35-s + 3.70·37-s − 2.38·38-s + 2.23·40-s − 0.472·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.447·5-s − 0.377·7-s + 0.790·8-s − 0.511·10-s + 0.372·11-s − 1.79·13-s + 0.432·14-s − 1.21·16-s − 0.0572·17-s + 0.337·19-s + 0.138·20-s − 0.426·22-s − 0.367·23-s + 0.200·25-s + 2.05·26-s − 0.116·28-s + 1.05·29-s − 1.61·31-s + 0.597·32-s + 0.0655·34-s − 0.169·35-s + 0.609·37-s − 0.386·38-s + 0.353·40-s − 0.0737·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 - 0.472T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 5.18T + 79T^{2} \) |
| 83 | \( 1 + 11T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−9.516101409374604278912134862481, −9.112057145407822315140618607633, −7.976752106883715955603529836923, −7.32263022971068144281479645382, −6.46909998263919254812697152610, −5.24523335528593048315957785465, −4.34918745977501108158803504976, −2.82483268527007168969748643369, −1.62609507910912674348794256331, 0,
1.62609507910912674348794256331, 2.82483268527007168969748643369, 4.34918745977501108158803504976, 5.24523335528593048315957785465, 6.46909998263919254812697152610, 7.32263022971068144281479645382, 7.976752106883715955603529836923, 9.112057145407822315140618607633, 9.516101409374604278912134862481
