| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 4·5-s − 3·6-s + 26·7-s − 15·8-s + 9·9-s − 4·10-s + 21·12-s + 32·13-s + 26·14-s + 12·15-s + 41·16-s − 74·17-s + 9·18-s + 60·19-s + 28·20-s − 78·21-s − 182·23-s + 45·24-s − 109·25-s + 32·26-s − 27·27-s − 182·28-s + 90·29-s + 12·30-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.357·5-s − 0.204·6-s + 1.40·7-s − 0.662·8-s + 1/3·9-s − 0.126·10-s + 0.505·12-s + 0.682·13-s + 0.496·14-s + 0.206·15-s + 0.640·16-s − 1.05·17-s + 0.117·18-s + 0.724·19-s + 0.313·20-s − 0.810·21-s − 1.64·23-s + 0.382·24-s − 0.871·25-s + 0.241·26-s − 0.192·27-s − 1.22·28-s + 0.576·29-s + 0.0730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 182 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 66 T + p^{3} T^{2} \) |
| 41 | \( 1 + 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 408 T + p^{3} T^{2} \) |
| 47 | \( 1 + 506 T + p^{3} T^{2} \) |
| 53 | \( 1 - 348 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 132 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 - 762 T + p^{3} T^{2} \) |
| 73 | \( 1 - 542 T + p^{3} T^{2} \) |
| 79 | \( 1 - 550 T + p^{3} T^{2} \) |
| 83 | \( 1 - 132 T + p^{3} T^{2} \) |
| 89 | \( 1 - 570 T + p^{3} T^{2} \) |
| 97 | \( 1 - 14 T + p^{3} T^{2} \) |
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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
−10.68986559669030505440389264907, −9.685135409559019949928867758367, −8.452522330878210111894997890577, −7.970854895944839855249943744404, −6.49448391590043741937216313361, −5.36671480335449753650669227405, −4.61367754936720743312896877246, −3.70160502417944553440467732930, −1.63346534628870380534544130601, 0,
1.63346534628870380534544130601, 3.70160502417944553440467732930, 4.61367754936720743312896877246, 5.36671480335449753650669227405, 6.49448391590043741937216313361, 7.970854895944839855249943744404, 8.452522330878210111894997890577, 9.685135409559019949928867758367, 10.68986559669030505440389264907
