| L(s) = 1 | − 1.94·2-s − 0.752·3-s + 1.78·4-s + 1.46·6-s + 1.49·7-s + 0.417·8-s − 2.43·9-s + 4.03·11-s − 1.34·12-s − 2.91·14-s − 4.38·16-s + 3.62·17-s + 4.73·18-s − 6.32·19-s − 1.12·21-s − 7.84·22-s + 0.957·23-s − 0.314·24-s + 4.08·27-s + 2.67·28-s + 1.88·29-s + 2.33·31-s + 7.69·32-s − 3.03·33-s − 7.05·34-s − 4.34·36-s − 11.2·37-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.434·3-s + 0.892·4-s + 0.597·6-s + 0.565·7-s + 0.147·8-s − 0.811·9-s + 1.21·11-s − 0.387·12-s − 0.777·14-s − 1.09·16-s + 0.879·17-s + 1.11·18-s − 1.45·19-s − 0.245·21-s − 1.67·22-s + 0.199·23-s − 0.0641·24-s + 0.786·27-s + 0.504·28-s + 0.350·29-s + 0.418·31-s + 1.35·32-s − 0.527·33-s − 1.21·34-s − 0.724·36-s − 1.84·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 3 | \( 1 + 0.752T + 3T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 0.957T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 - 6.85T + 47T^{2} \) |
| 53 | \( 1 + 3.88T + 53T^{2} \) |
| 59 | \( 1 + 5.27T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 2.55T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 + 8.98T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.256035600624595027327452117332, −7.49673614641707378040325815301, −6.66290782202269998333298697039, −6.11170890236615869759307720286, −5.08092950542114870981867683730, −4.33326080729965124563599559949, −3.23497513832295064161912462987, −1.98241264647123904647264398865, −1.18911262590771112740047617483, 0,
1.18911262590771112740047617483, 1.98241264647123904647264398865, 3.23497513832295064161912462987, 4.33326080729965124563599559949, 5.08092950542114870981867683730, 6.11170890236615869759307720286, 6.66290782202269998333298697039, 7.49673614641707378040325815301, 8.256035600624595027327452117332
