| L(s) = 1 | − 3.53·2-s − 3·3-s + 4.46·4-s − 5·5-s + 10.5·6-s + 12.4·8-s + 9·9-s + 17.6·10-s − 2.93·11-s − 13.4·12-s + 19.0·13-s + 15·15-s − 79.7·16-s − 122.·17-s − 31.7·18-s − 107.·19-s − 22.3·20-s + 10.3·22-s + 210.·23-s − 37.4·24-s + 25·25-s − 67.3·26-s − 27·27-s + 95.4·29-s − 52.9·30-s + 94.3·31-s + 181.·32-s + ⋯ |
| L(s) = 1 | − 1.24·2-s − 0.577·3-s + 0.558·4-s − 0.447·5-s + 0.720·6-s + 0.551·8-s + 0.333·9-s + 0.558·10-s − 0.0805·11-s − 0.322·12-s + 0.406·13-s + 0.258·15-s − 1.24·16-s − 1.74·17-s − 0.416·18-s − 1.29·19-s − 0.249·20-s + 0.100·22-s + 1.90·23-s − 0.318·24-s + 0.200·25-s − 0.507·26-s − 0.192·27-s + 0.611·29-s − 0.322·30-s + 0.546·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.53T + 8T^{2} \) |
| 11 | \( 1 + 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 43.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−9.411982426372493529729514189457, −8.745001715394745214491684725138, −8.065227875334943420692308503853, −6.95963816708906646539237970173, −6.43239991955889644454996613334, −4.89056365480134540174576408687, −4.18131232564101398669639312479, −2.45221579180846618649857924098, −1.05497619576561492344049950820, 0,
1.05497619576561492344049950820, 2.45221579180846618649857924098, 4.18131232564101398669639312479, 4.89056365480134540174576408687, 6.43239991955889644454996613334, 6.95963816708906646539237970173, 8.065227875334943420692308503853, 8.745001715394745214491684725138, 9.411982426372493529729514189457
