| L(s) = 1 | + 4.12·2-s + 158.·3-s − 494.·4-s + 522.·5-s + 652.·6-s − 9.36e3·7-s − 4.15e3·8-s + 5.37e3·9-s + 2.15e3·10-s − 6.95e3·11-s − 7.83e4·12-s − 7.74e4·13-s − 3.86e4·14-s + 8.27e4·15-s + 2.36e5·16-s + 2.73e3·17-s + 2.21e4·18-s − 8.17e4·19-s − 2.58e5·20-s − 1.48e6·21-s − 2.86e4·22-s − 1.95e6·23-s − 6.57e5·24-s − 1.67e6·25-s − 3.19e5·26-s − 2.26e6·27-s + 4.63e6·28-s + ⋯ |
| L(s) = 1 | + 0.182·2-s + 1.12·3-s − 0.966·4-s + 0.374·5-s + 0.205·6-s − 1.47·7-s − 0.358·8-s + 0.273·9-s + 0.0681·10-s − 0.143·11-s − 1.09·12-s − 0.751·13-s − 0.268·14-s + 0.422·15-s + 0.901·16-s + 0.00793·17-s + 0.0497·18-s − 0.143·19-s − 0.361·20-s − 1.66·21-s − 0.0261·22-s − 1.45·23-s − 0.404·24-s − 0.860·25-s − 0.136·26-s − 0.820·27-s + 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 7.07e5T \) |
| good | 2 | \( 1 - 4.12T + 512T^{2} \) |
| 3 | \( 1 - 158.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 522.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.36e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.95e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.74e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.73e3T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.17e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.95e6T + 1.80e12T^{2} \) |
| 31 | \( 1 - 2.30e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.66e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.60e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.61e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.64e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.70e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.62e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.99e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.04e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.61e9T + 7.60e17T^{2} \) |
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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
−14.16015889362053297696450895601, −13.43515336341281757411079575290, −12.41750486799236375211181589243, −9.887887974143674865161752381407, −9.329225721969301865909837654129, −7.925362581710914910941599138131, −5.95862714918613253106614170253, −3.95068650521229364936483425148, −2.63444628042328978558831269969, 0,
2.63444628042328978558831269969, 3.95068650521229364936483425148, 5.95862714918613253106614170253, 7.925362581710914910941599138131, 9.329225721969301865909837654129, 9.887887974143674865161752381407, 12.41750486799236375211181589243, 13.43515336341281757411079575290, 14.16015889362053297696450895601
