| L(s) = 1 | − 16·2-s + 81·3-s + 256·4-s + 76·5-s − 1.29e3·6-s − 4.09e3·8-s + 6.56e3·9-s − 1.21e3·10-s + 3.83e4·11-s + 2.07e4·12-s − 9.82e4·13-s + 6.15e3·15-s + 6.55e4·16-s − 1.04e5·17-s − 1.04e5·18-s + 4.20e5·19-s + 1.94e4·20-s − 6.14e5·22-s − 1.39e5·23-s − 3.31e5·24-s − 1.94e6·25-s + 1.57e6·26-s + 5.31e5·27-s − 1.91e6·29-s − 9.84e4·30-s + 6.37e6·31-s − 1.04e6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.0543·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.0384·10-s + 0.790·11-s + 0.288·12-s − 0.954·13-s + 0.0313·15-s + 1/4·16-s − 0.303·17-s − 0.235·18-s + 0.740·19-s + 0.0271·20-s − 0.558·22-s − 0.103·23-s − 0.204·24-s − 0.997·25-s + 0.674·26-s + 0.192·27-s − 0.503·29-s − 0.0222·30-s + 1.24·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 - p^{4} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 76 T + p^{9} T^{2} \) |
| 11 | \( 1 - 38386 T + p^{9} T^{2} \) |
| 13 | \( 1 + 98298 T + p^{9} T^{2} \) |
| 17 | \( 1 + 104524 T + p^{9} T^{2} \) |
| 19 | \( 1 - 420580 T + p^{9} T^{2} \) |
| 23 | \( 1 + 139118 T + p^{9} T^{2} \) |
| 29 | \( 1 + 1916290 T + p^{9} T^{2} \) |
| 31 | \( 1 - 6379488 T + p^{9} T^{2} \) |
| 37 | \( 1 + 6629278 T + p^{9} T^{2} \) |
| 41 | \( 1 - 6692112 T + p^{9} T^{2} \) |
| 43 | \( 1 + 23269732 T + p^{9} T^{2} \) |
| 47 | \( 1 - 22000596 T + p^{9} T^{2} \) |
| 53 | \( 1 - 18919770 T + p^{9} T^{2} \) |
| 59 | \( 1 + 179035544 T + p^{9} T^{2} \) |
| 61 | \( 1 - 19797786 T + p^{9} T^{2} \) |
| 67 | \( 1 + 263015240 T + p^{9} T^{2} \) |
| 71 | \( 1 - 22447678 T + p^{9} T^{2} \) |
| 73 | \( 1 + 11023774 T + p^{9} T^{2} \) |
| 79 | \( 1 + 284917908 T + p^{9} T^{2} \) |
| 83 | \( 1 - 226865924 T + p^{9} T^{2} \) |
| 89 | \( 1 - 191377296 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1162236578 T + p^{9} T^{2} \) |
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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
−9.605083953352980686852875469022, −8.914307098434668118560959751556, −7.85994526955503223168915842830, −7.11604565596641529339698579706, −6.03210364474251277098783725357, −4.63910782466909243693851030773, −3.40538091158149579778656389712, −2.29867280478219999489315965007, −1.29704622966914576993545882125, 0,
1.29704622966914576993545882125, 2.29867280478219999489315965007, 3.40538091158149579778656389712, 4.63910782466909243693851030773, 6.03210364474251277098783725357, 7.11604565596641529339698579706, 7.85994526955503223168915842830, 8.914307098434668118560959751556, 9.605083953352980686852875469022
