| L(s) = 1 | + 1.43·2-s − 3-s + 0.0686·4-s − 5-s − 1.43·6-s − 2.74·7-s − 2.77·8-s + 9-s − 1.43·10-s + 2.74·11-s − 0.0686·12-s − 5.14·13-s − 3.94·14-s + 15-s − 4.13·16-s − 3.72·17-s + 1.43·18-s − 0.404·19-s − 0.0686·20-s + 2.74·21-s + 3.94·22-s − 5.45·23-s + 2.77·24-s + 25-s − 7.40·26-s − 27-s − 0.188·28-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.577·3-s + 0.0343·4-s − 0.447·5-s − 0.587·6-s − 1.03·7-s − 0.982·8-s + 0.333·9-s − 0.454·10-s + 0.827·11-s − 0.0198·12-s − 1.42·13-s − 1.05·14-s + 0.258·15-s − 1.03·16-s − 0.904·17-s + 0.339·18-s − 0.0927·19-s − 0.0153·20-s + 0.598·21-s + 0.841·22-s − 1.13·23-s + 0.567·24-s + 0.200·25-s − 1.45·26-s − 0.192·27-s − 0.0355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 19 | \( 1 + 0.404T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 31 | \( 1 - 1.45T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 6.43T + 53T^{2} \) |
| 59 | \( 1 + 9.91T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 1.62T + 83T^{2} \) |
| 89 | \( 1 + 8.87T + 89T^{2} \) |
| 97 | \( 1 - 7.82T + 97T^{2} \) |
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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
−10.95656062302615631410805329783, −9.686553294177433040563724238810, −9.182258842225644903130252942526, −7.66660610259077591111994634778, −6.55467192115314912036789786229, −5.94610588804771616400788435859, −4.64370520967590416973420014453, −4.01526746204417431489995931593, −2.70587088047337827461922603862, 0,
2.70587088047337827461922603862, 4.01526746204417431489995931593, 4.64370520967590416973420014453, 5.94610588804771616400788435859, 6.55467192115314912036789786229, 7.66660610259077591111994634778, 9.182258842225644903130252942526, 9.686553294177433040563724238810, 10.95656062302615631410805329783
