| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 6·5-s − 36·6-s + 64·8-s + 81·9-s + 24·10-s − 666·11-s − 144·12-s + 559·13-s − 54·15-s + 256·16-s + 1.74e3·17-s + 324·18-s − 1.15e3·19-s + 96·20-s − 2.66e3·22-s − 3.46e3·23-s − 576·24-s − 3.08e3·25-s + 2.23e3·26-s − 729·27-s + 3.37e3·29-s − 216·30-s − 6.29e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.107·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.0758·10-s − 1.65·11-s − 0.288·12-s + 0.917·13-s − 0.0619·15-s + 1/4·16-s + 1.46·17-s + 0.235·18-s − 0.735·19-s + 0.0536·20-s − 1.17·22-s − 1.36·23-s − 0.204·24-s − 0.988·25-s + 0.648·26-s − 0.192·27-s + 0.744·29-s − 0.0438·30-s − 1.17·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 11 | \( 1 + 666 T + p^{5} T^{2} \) |
| 13 | \( 1 - 43 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1740 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1157 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3468 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3372 T + p^{5} T^{2} \) |
| 31 | \( 1 + 203 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 3131 T + p^{5} T^{2} \) |
| 41 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11407 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2310 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28296 T + p^{5} T^{2} \) |
| 59 | \( 1 + 20544 T + p^{5} T^{2} \) |
| 61 | \( 1 - 4630 T + p^{5} T^{2} \) |
| 67 | \( 1 + 18745 T + p^{5} T^{2} \) |
| 71 | \( 1 + 38226 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70589 T + p^{5} T^{2} \) |
| 79 | \( 1 + 62293 T + p^{5} T^{2} \) |
| 83 | \( 1 + 79818 T + p^{5} T^{2} \) |
| 89 | \( 1 - 18120 T + p^{5} T^{2} \) |
| 97 | \( 1 + 124754 T + p^{5} T^{2} \) |
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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
−10.54993920027823863704108978803, −9.931304251034525736123619334239, −8.254027796626366659307480027620, −7.50177768939488291446826229838, −6.06663401348382362301575337234, −5.57443980472509031880262589118, −4.37498243713819438479013534453, −3.13832296735515526073587324644, −1.70962022664517275485913801710, 0,
1.70962022664517275485913801710, 3.13832296735515526073587324644, 4.37498243713819438479013534453, 5.57443980472509031880262589118, 6.06663401348382362301575337234, 7.50177768939488291446826229838, 8.254027796626366659307480027620, 9.931304251034525736123619334239, 10.54993920027823863704108978803
