| L(s) = 1 | − 2·4-s − 3.46·5-s + 1.73·7-s + 3.46·11-s + 4·16-s + 3.46·19-s + 6.92·20-s − 6·23-s + 6.99·25-s − 3.46·28-s − 6·29-s + 1.73·31-s − 5.99·35-s − 6.92·41-s + 43-s − 6.92·44-s + 3.46·47-s − 4·49-s − 12·53-s − 11.9·55-s − 3.46·59-s + 61-s − 8·64-s − 8.66·67-s + 10.3·71-s − 1.73·73-s − 6.92·76-s + ⋯ |
| L(s) = 1 | − 4-s − 1.54·5-s + 0.654·7-s + 1.04·11-s + 16-s + 0.794·19-s + 1.54·20-s − 1.25·23-s + 1.39·25-s − 0.654·28-s − 1.11·29-s + 0.311·31-s − 1.01·35-s − 1.08·41-s + 0.152·43-s − 1.04·44-s + 0.505·47-s − 0.571·49-s − 1.64·53-s − 1.61·55-s − 0.450·59-s + 0.128·61-s − 64-s − 1.05·67-s + 1.23·71-s − 0.202·73-s − 0.794·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
| 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
−9.023759865092393766925130882852, −8.050875138516453875748684461194, −7.86295233529838917236644390129, −6.78191667369722655303971926722, −5.58955418777624952506230146765, −4.62565579207585428587006442781, −4.01314425454951976637895234458, −3.34122237006366957014266780101, −1.41433462402830073203810317989, 0,
1.41433462402830073203810317989, 3.34122237006366957014266780101, 4.01314425454951976637895234458, 4.62565579207585428587006442781, 5.58955418777624952506230146765, 6.78191667369722655303971926722, 7.86295233529838917236644390129, 8.050875138516453875748684461194, 9.023759865092393766925130882852
