L(s) = 1 | − 2.65·2-s + 5.04·4-s − 0.121·5-s − 0.0602·7-s − 8.08·8-s + 0.323·10-s + 0.159·11-s − 1.06·13-s + 0.159·14-s + 11.3·16-s + 3.67·17-s − 2.41·19-s − 0.614·20-s − 0.424·22-s − 3.62·23-s − 4.98·25-s + 2.81·26-s − 0.303·28-s − 5.54·29-s + 4.40·31-s − 14.0·32-s − 9.74·34-s + 0.00734·35-s − 7.01·37-s + 6.41·38-s + 0.985·40-s + 9.62·41-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 2.52·4-s − 0.0545·5-s − 0.0227·7-s − 2.85·8-s + 0.102·10-s + 0.0482·11-s − 0.294·13-s + 0.0427·14-s + 2.84·16-s + 0.890·17-s − 0.554·19-s − 0.137·20-s − 0.0905·22-s − 0.756·23-s − 0.997·25-s + 0.551·26-s − 0.0574·28-s − 1.02·29-s + 0.791·31-s − 2.47·32-s − 1.67·34-s + 0.00124·35-s − 1.15·37-s + 1.04·38-s + 0.155·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 + 0.121T + 5T^{2} \) |
| 7 | \( 1 + 0.0602T + 7T^{2} \) |
| 11 | \( 1 - 0.159T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 3.67T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 3.62T + 23T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 - 4.40T + 31T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 - 9.62T + 41T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 - 0.250T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 8.57T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 7.67T + 79T^{2} \) |
| 83 | \( 1 + 7.42T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 0.974T + 97T^{2} \) |
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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
−9.573530829935091108424834207971, −8.505032033922729586246856745006, −7.989537151027761859253959254746, −7.22438743357469619940664263344, −6.38695994345498654932656467692, −5.47699039765571605413702617627, −3.81419502257875743152068374570, −2.53778388717430804672205373606, −1.50458402205591738287703042178, 0,
1.50458402205591738287703042178, 2.53778388717430804672205373606, 3.81419502257875743152068374570, 5.47699039765571605413702617627, 6.38695994345498654932656467692, 7.22438743357469619940664263344, 7.989537151027761859253959254746, 8.505032033922729586246856745006, 9.573530829935091108424834207971