| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 2·11-s + 2·12-s + 15-s + 4·16-s + 17-s + 2·19-s + 2·20-s − 21-s − 3·23-s + 25-s − 27-s − 2·28-s + 4·29-s + 5·31-s − 2·33-s − 35-s − 2·36-s + 37-s − 11·41-s + 4·43-s − 4·44-s − 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 0.258·15-s + 16-s + 0.242·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.742·29-s + 0.898·31-s − 0.348·33-s − 0.169·35-s − 1/3·36-s + 0.164·37-s − 1.71·41-s + 0.609·43-s − 0.603·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.89877144331652, −12.27962730735493, −11.99328837657406, −11.72664078083644, −11.13005736428082, −10.58453113843276, −10.08523307195714, −9.829144715547709, −9.244899986329830, −8.625582500719730, −8.428217503745369, −7.844515158471888, −7.368602958370576, −6.856860557575194, −6.225629587704362, −5.848385377721824, −5.182172350555877, −4.837041763004639, −4.409202831071767, −3.691204189035641, −3.602862307571243, −2.669846499508644, −1.977242003348914, −1.063274832796751, −0.8596194942138349, 0,
0.8596194942138349, 1.063274832796751, 1.977242003348914, 2.669846499508644, 3.602862307571243, 3.691204189035641, 4.409202831071767, 4.837041763004639, 5.182172350555877, 5.848385377721824, 6.225629587704362, 6.856860557575194, 7.368602958370576, 7.844515158471888, 8.428217503745369, 8.625582500719730, 9.244899986329830, 9.829144715547709, 10.08523307195714, 10.58453113843276, 11.13005736428082, 11.72664078083644, 11.99328837657406, 12.27962730735493, 12.89877144331652
