| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s − 11-s + 2·12-s + 4·13-s + 14-s + 16-s − 18-s − 4·19-s − 2·21-s + 22-s − 2·24-s − 4·26-s − 4·27-s − 28-s − 6·29-s − 10·31-s − 32-s − 2·33-s + 36-s − 2·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.436·21-s + 0.213·22-s − 0.408·24-s − 0.784·26-s − 0.769·27-s − 0.188·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.251404203517565513991144425922, −7.64643700459544933868049425842, −6.84225959024603351556200506453, −6.07448813870957861281114248645, −5.21563492662287784876169064504, −3.78202223483943311115680668239, −3.45066066607841924172297512404, −2.34921844858892087621262850049, −1.63539925073102373815460642490, 0,
1.63539925073102373815460642490, 2.34921844858892087621262850049, 3.45066066607841924172297512404, 3.78202223483943311115680668239, 5.21563492662287784876169064504, 6.07448813870957861281114248645, 6.84225959024603351556200506453, 7.64643700459544933868049425842, 8.251404203517565513991144425922
