| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 2·11-s − 12-s + 14-s + 16-s + 3·17-s − 18-s − 8·19-s + 21-s + 2·22-s + 4·23-s + 24-s − 27-s − 28-s + 9·29-s + 4·31-s − 32-s + 2·33-s − 3·34-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.83·19-s + 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s + 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.348·33-s − 0.514·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376325496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376325496\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 1063 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−13.20547473707504, −12.55388384634882, −12.32368295152815, −12.00627777865821, −11.04895598406863, −10.86262481398643, −10.53910813391042, −9.992190544261474, −9.447310247626747, −9.064933805039595, −8.397054474495984, −7.946673922784902, −7.617065602543549, −6.779860492840206, −6.484469646862562, −6.100768531641856, −5.439084455804656, −4.844944790917833, −4.347076083830042, −3.709299029891476, −2.817907482732138, −2.586360607967455, −1.770819519132704, −0.9006297709506010, −0.5119170620021974,
0.5119170620021974, 0.9006297709506010, 1.770819519132704, 2.586360607967455, 2.817907482732138, 3.709299029891476, 4.347076083830042, 4.844944790917833, 5.439084455804656, 6.100768531641856, 6.484469646862562, 6.779860492840206, 7.617065602543549, 7.946673922784902, 8.397054474495984, 9.064933805039595, 9.447310247626747, 9.992190544261474, 10.53910813391042, 10.86262481398643, 11.04895598406863, 12.00627777865821, 12.32368295152815, 12.55388384634882, 13.20547473707504
