| L(s) = 1 | − 3.07·2-s + 3·3-s + 1.48·4-s − 5·5-s − 9.23·6-s + 23.1·7-s + 20.0·8-s + 9·9-s + 15.3·10-s − 43.1·11-s + 4.44·12-s − 74.6·13-s − 71.2·14-s − 15·15-s − 73.6·16-s + 82.7·17-s − 27.7·18-s − 88.2·19-s − 7.40·20-s + 69.4·21-s + 132.·22-s + 175.·23-s + 60.2·24-s + 25·25-s + 229.·26-s + 27·27-s + 34.3·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.185·4-s − 0.447·5-s − 0.628·6-s + 1.25·7-s + 0.887·8-s + 0.333·9-s + 0.486·10-s − 1.18·11-s + 0.106·12-s − 1.59·13-s − 1.36·14-s − 0.258·15-s − 1.15·16-s + 1.18·17-s − 0.362·18-s − 1.06·19-s − 0.0828·20-s + 0.721·21-s + 1.28·22-s + 1.59·23-s + 0.512·24-s + 0.200·25-s + 1.73·26-s + 0.192·27-s + 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 3.07T + 8T^{2} \) |
| 7 | \( 1 - 23.1T + 343T^{2} \) |
| 11 | \( 1 + 43.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 74.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 14.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 265.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 645.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 386.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 737.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 714.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 518.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 946.T + 9.12e5T^{2} \) |
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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
−10.29749038920076942746237463422, −9.205323099334341716771045561433, −8.404929803328449112433015438000, −7.68021141501735037139878073975, −7.27546236379277083526627012568, −5.15550088687433650294532513695, −4.54174264830510166633204995829, −2.80681374297313137501342389484, −1.54643358343931661440859537037, 0,
1.54643358343931661440859537037, 2.80681374297313137501342389484, 4.54174264830510166633204995829, 5.15550088687433650294532513695, 7.27546236379277083526627012568, 7.68021141501735037139878073975, 8.404929803328449112433015438000, 9.205323099334341716771045561433, 10.29749038920076942746237463422
