L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 6·11-s − 12-s − 13-s − 15-s + 16-s + 4·17-s − 18-s + 20-s − 6·22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 9·29-s + 30-s − 32-s − 6·33-s − 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s − 1.27·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.67·29-s + 0.182·30-s − 0.176·32-s − 1.04·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.308231292\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.308231292\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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
−13.40256947637439, −12.62958274259332, −12.44038465125193, −11.88262516575419, −11.46264868802900, −10.94589847944153, −10.61111290660088, −9.878675552513847, −9.502431313497121, −9.213509720379188, −8.727200609848461, −8.007224976723336, −7.425624737300576, −7.041615348380836, −6.567149827478031, −5.873330951839022, −5.723914287938796, −4.975827971035390, −4.276706038708641, −3.686717586061944, −3.230202325717072, −2.227091398446106, −1.794968485406161, −1.042673425538495, −0.6293706391732628,
0.6293706391732628, 1.042673425538495, 1.794968485406161, 2.227091398446106, 3.230202325717072, 3.686717586061944, 4.276706038708641, 4.975827971035390, 5.723914287938796, 5.873330951839022, 6.567149827478031, 7.041615348380836, 7.425624737300576, 8.007224976723336, 8.727200609848461, 9.213509720379188, 9.502431313497121, 9.878675552513847, 10.61111290660088, 10.94589847944153, 11.46264868802900, 11.88262516575419, 12.44038465125193, 12.62958274259332, 13.40256947637439