| L(s) = 1 | + 2-s − 3-s + 4-s + 3.23·5-s − 6-s + 8-s + 9-s + 3.23·10-s − 0.763·11-s − 12-s + 4.47·13-s − 3.23·15-s + 16-s + 4·17-s + 18-s + 7.70·19-s + 3.23·20-s − 0.763·22-s + 23-s − 24-s + 5.47·25-s + 4.47·26-s − 27-s + 4.47·29-s − 3.23·30-s − 6.47·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.44·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.02·10-s − 0.230·11-s − 0.288·12-s + 1.24·13-s − 0.835·15-s + 0.250·16-s + 0.970·17-s + 0.235·18-s + 1.76·19-s + 0.723·20-s − 0.162·22-s + 0.208·23-s − 0.204·24-s + 1.09·25-s + 0.877·26-s − 0.192·27-s + 0.830·29-s − 0.590·30-s − 1.16·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.333426765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.333426765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9.23T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 0.763T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
| 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
−7.74411312925280007893831169427, −7.11598081722120877698371690495, −6.12880574620820230116965418109, −5.93640054438167707485972734348, −5.27470456384454945037502439596, −4.64647748014847667139688059679, −3.46807072122014520108131867037, −2.91926655307911333168944879329, −1.68827362115027907319905257884, −1.11892676481292312876691637756,
1.11892676481292312876691637756, 1.68827362115027907319905257884, 2.91926655307911333168944879329, 3.46807072122014520108131867037, 4.64647748014847667139688059679, 5.27470456384454945037502439596, 5.93640054438167707485972734348, 6.12880574620820230116965418109, 7.11598081722120877698371690495, 7.74411312925280007893831169427
