| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 3.06·13-s + 14-s + 16-s − 5.06·17-s − 3.38·19-s − 20-s − 22-s + 8.58·23-s + 25-s − 3.06·26-s + 28-s + 4.12·29-s + 1.38·31-s + 32-s − 5.06·34-s − 35-s + 1.06·37-s − 3.38·38-s − 40-s − 0.610·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.849·13-s + 0.267·14-s + 0.250·16-s − 1.22·17-s − 0.777·19-s − 0.223·20-s − 0.213·22-s + 1.78·23-s + 0.200·25-s − 0.600·26-s + 0.188·28-s + 0.766·29-s + 0.249·31-s + 0.176·32-s − 0.868·34-s − 0.169·35-s + 0.174·37-s − 0.549·38-s − 0.158·40-s − 0.0953·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 - 1.38T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 + 0.610T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
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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
−7.46058247369271883884266781880, −6.83819790542274522153517022095, −6.30561793755052304715127144252, −5.20482311824732054480007586886, −4.72969424290869985291037188394, −4.20882470592292020496528421962, −3.08366971059549318394839482478, −2.53230337078197413757607874568, −1.46180583769146282861450246261, 0,
1.46180583769146282861450246261, 2.53230337078197413757607874568, 3.08366971059549318394839482478, 4.20882470592292020496528421962, 4.72969424290869985291037188394, 5.20482311824732054480007586886, 6.30561793755052304715127144252, 6.83819790542274522153517022095, 7.46058247369271883884266781880
