L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 12-s + 6·13-s − 2·14-s − 15-s + 16-s − 6·17-s − 18-s − 2·19-s − 20-s + 2·21-s + 6·23-s − 24-s + 25-s − 6·26-s + 27-s + 2·28-s + 7·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.436·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.377·28-s + 1.29·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.811843971\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.811843971\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.39004381283149, −13.08637141914850, −12.51001357574809, −11.81563399919307, −11.41064640997801, −10.96085186327771, −10.65619771305003, −10.16094301632057, −9.381028006207811, −8.791847377846960, −8.619142690045432, −8.354097736842065, −7.641371146265283, −7.156001908768891, −6.618817878454307, −6.209416663953169, −5.493172998026504, −4.704452023654043, −4.289963979246743, −3.784148537741325, −2.966999923525443, −2.576362501538170, −1.772269400456885, −1.210386234747023, −0.5925645107421178,
0.5925645107421178, 1.210386234747023, 1.772269400456885, 2.576362501538170, 2.966999923525443, 3.784148537741325, 4.289963979246743, 4.704452023654043, 5.493172998026504, 6.209416663953169, 6.618817878454307, 7.156001908768891, 7.641371146265283, 8.354097736842065, 8.619142690045432, 8.791847377846960, 9.381028006207811, 10.16094301632057, 10.65619771305003, 10.96085186327771, 11.41064640997801, 11.81563399919307, 12.51001357574809, 13.08637141914850, 13.39004381283149