| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s + 8-s + 9-s − 2·10-s − 6·11-s − 2·12-s + 4·13-s + 4·15-s + 16-s − 2·17-s + 18-s + 4·19-s − 2·20-s − 6·22-s − 2·24-s − 25-s + 4·26-s + 4·27-s − 10·29-s + 4·30-s + 8·31-s + 32-s + 12·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 1.27·22-s − 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 1.85·29-s + 0.730·30-s + 1.43·31-s + 0.176·32-s + 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7867885821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7867885821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.53864958113600, −13.76232024869994, −13.34375007664410, −12.94920707035157, −12.46668500870717, −11.79009571903346, −11.38148549663740, −11.05539048449903, −10.78447158940689, −9.924496859717641, −9.534977517799437, −8.424763406416450, −8.033428163820863, −7.679947820271432, −6.917917397186750, −6.326306942699239, −5.844152944344007, −5.294945015151253, −4.850962011064751, −4.290184081250471, −3.491322596325512, −3.045687076519418, −2.227221228893342, −1.236526063541678, −0.3129831563941714,
0.3129831563941714, 1.236526063541678, 2.227221228893342, 3.045687076519418, 3.491322596325512, 4.290184081250471, 4.850962011064751, 5.294945015151253, 5.844152944344007, 6.326306942699239, 6.917917397186750, 7.679947820271432, 8.033428163820863, 8.424763406416450, 9.534977517799437, 9.924496859717641, 10.78447158940689, 11.05539048449903, 11.38148549663740, 11.79009571903346, 12.46668500870717, 12.94920707035157, 13.34375007664410, 13.76232024869994, 14.53864958113600
