| L(s) = 1 | − 4.06·2-s + 8.49·4-s − 16.2·5-s + 6.77·7-s − 2.00·8-s + 66.0·10-s + 16.5·11-s − 82.3·13-s − 27.4·14-s − 59.7·16-s − 6.48·17-s + 151.·19-s − 138.·20-s − 67.3·22-s + 154.·23-s + 139.·25-s + 334.·26-s + 57.5·28-s − 220.·29-s − 79.4·31-s + 258.·32-s + 26.3·34-s − 110.·35-s + 414.·37-s − 613.·38-s + 32.7·40-s + 521.·41-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.06·4-s − 1.45·5-s + 0.365·7-s − 0.0888·8-s + 2.08·10-s + 0.454·11-s − 1.75·13-s − 0.524·14-s − 0.934·16-s − 0.0924·17-s + 1.82·19-s − 1.54·20-s − 0.652·22-s + 1.39·23-s + 1.11·25-s + 2.52·26-s + 0.388·28-s − 1.41·29-s − 0.460·31-s + 1.43·32-s + 0.132·34-s − 0.532·35-s + 1.83·37-s − 2.61·38-s + 0.129·40-s + 1.98·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 4.06T + 8T^{2} \) |
| 5 | \( 1 + 16.2T + 125T^{2} \) |
| 7 | \( 1 - 6.77T + 343T^{2} \) |
| 11 | \( 1 - 16.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.48T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 414.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 521.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 265.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 31.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 256.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 68.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 725.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 571.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 304.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 731.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 694.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 597.T + 9.12e5T^{2} \) |
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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
−9.586295432050511425246654741678, −9.314393440148277004147314499633, −8.052403512468917090194987940615, −7.51262373433037350781988776050, −7.05751104895759179630570749805, −5.18689431995158924538975172147, −4.18531264390581627639922329763, −2.77293161098008167621132044857, −1.12778302726298450038598506842, 0,
1.12778302726298450038598506842, 2.77293161098008167621132044857, 4.18531264390581627639922329763, 5.18689431995158924538975172147, 7.05751104895759179630570749805, 7.51262373433037350781988776050, 8.052403512468917090194987940615, 9.314393440148277004147314499633, 9.586295432050511425246654741678
