| L(s) = 1 | + 0.0532·2-s − 1.99·4-s + 3.85·5-s + 4.59·7-s − 0.212·8-s + 0.205·10-s − 2.84·11-s + 5.18·13-s + 0.244·14-s + 3.98·16-s + 3.00·17-s − 1.00·19-s − 7.69·20-s − 0.151·22-s − 6.96·23-s + 9.86·25-s + 0.276·26-s − 9.16·28-s − 5.04·29-s − 8.71·31-s + 0.638·32-s + 0.160·34-s + 17.6·35-s − 0.247·37-s − 0.0537·38-s − 0.821·40-s + 9.02·41-s + ⋯ |
| L(s) = 1 | + 0.0376·2-s − 0.998·4-s + 1.72·5-s + 1.73·7-s − 0.0753·8-s + 0.0649·10-s − 0.857·11-s + 1.43·13-s + 0.0653·14-s + 0.995·16-s + 0.729·17-s − 0.231·19-s − 1.72·20-s − 0.0323·22-s − 1.45·23-s + 1.97·25-s + 0.0542·26-s − 1.73·28-s − 0.936·29-s − 1.56·31-s + 0.112·32-s + 0.0274·34-s + 2.99·35-s − 0.0407·37-s − 0.00871·38-s − 0.129·40-s + 1.41·41-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
| \(L(1)\) |
\(\approx\) |
\(2.202779914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.202779914\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.0532T + 2T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 + 5.04T + 29T^{2} \) |
| 31 | \( 1 + 8.71T + 31T^{2} \) |
| 37 | \( 1 + 0.247T + 37T^{2} \) |
| 41 | \( 1 - 9.02T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 0.912T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 0.554T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 + 4.63T + 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−9.685920930842159776959710525054, −9.056782328653822677054959636055, −8.234592280836038678557370577232, −7.63559650015852125866071579266, −5.90515705872799817414125163322, −5.66111532310115944760472383517, −4.82132634186669217751048931876, −3.76774102936385511920047500736, −2.15172608321577608712901138539, −1.31445630562917647044123863133,
1.31445630562917647044123863133, 2.15172608321577608712901138539, 3.76774102936385511920047500736, 4.82132634186669217751048931876, 5.66111532310115944760472383517, 5.90515705872799817414125163322, 7.63559650015852125866071579266, 8.234592280836038678557370577232, 9.056782328653822677054959636055, 9.685920930842159776959710525054
