| L(s) = 1 | + 2-s − 3-s + 4-s − 3.85·5-s − 6-s + 2.38·7-s + 8-s + 9-s − 3.85·10-s − 12-s + 1.23·13-s + 2.38·14-s + 3.85·15-s + 16-s + 6.47·17-s + 18-s + 1.23·19-s − 3.85·20-s − 2.38·21-s + 1.23·23-s − 24-s + 9.85·25-s + 1.23·26-s − 27-s + 2.38·28-s − 2.61·29-s + 3.85·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.72·5-s − 0.408·6-s + 0.900·7-s + 0.353·8-s + 0.333·9-s − 1.21·10-s − 0.288·12-s + 0.342·13-s + 0.636·14-s + 0.995·15-s + 0.250·16-s + 1.56·17-s + 0.235·18-s + 0.283·19-s − 0.861·20-s − 0.519·21-s + 0.257·23-s − 0.204·24-s + 1.97·25-s + 0.242·26-s − 0.192·27-s + 0.450·28-s − 0.486·29-s + 0.703·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684451205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684451205\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 4.32T + 97T^{2} \) |
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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
−10.92067197125289586043572238048, −9.721814621596484479229304074158, −8.175682440831836080633324142219, −7.84888020539591776689809252097, −6.93745081278728697200728645754, −5.75210121623670444513090577016, −4.80403804803836172190251838443, −4.08432534559556534644944218418, −3.09462095029405092696077743856, −1.08683083231975399393592624583,
1.08683083231975399393592624583, 3.09462095029405092696077743856, 4.08432534559556534644944218418, 4.80403804803836172190251838443, 5.75210121623670444513090577016, 6.93745081278728697200728645754, 7.84888020539591776689809252097, 8.175682440831836080633324142219, 9.721814621596484479229304074158, 10.92067197125289586043572238048
