L(s) = 1 | − 2.52·2-s − 0.792·3-s + 4.37·4-s + 2·6-s + 3.46·7-s − 5.98·8-s − 2.37·9-s − 11-s − 3.46·12-s − 8.74·14-s + 6.37·16-s + 1.58·17-s + 5.98·18-s + 4·19-s − 2.74·21-s + 2.52·22-s + 0.792·23-s + 4.74·24-s + 4.25·27-s + 15.1·28-s + 8.74·29-s + 3.37·31-s − 4.10·32-s + 0.792·33-s − 4·34-s − 10.3·36-s − 1.08·37-s + ⋯ |
L(s) = 1 | − 1.78·2-s − 0.457·3-s + 2.18·4-s + 0.816·6-s + 1.30·7-s − 2.11·8-s − 0.790·9-s − 0.301·11-s − 1.00·12-s − 2.33·14-s + 1.59·16-s + 0.384·17-s + 1.41·18-s + 0.917·19-s − 0.598·21-s + 0.538·22-s + 0.165·23-s + 0.968·24-s + 0.819·27-s + 2.86·28-s + 1.62·29-s + 0.605·31-s − 0.726·32-s + 0.137·33-s − 0.685·34-s − 1.72·36-s − 0.178·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5337035026\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5337035026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 + 0.792T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 0.792T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 7.37T + 59T^{2} \) |
| 61 | \( 1 + 0.744T + 61T^{2} \) |
| 67 | \( 1 - 9.30T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 - 5.84T + 97T^{2} \) |
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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
−11.52467847100609427449141012181, −10.83919911327928114927134262375, −10.03384997744713630229379197811, −8.885749534287500360974957337618, −8.163877458247526013046625358900, −7.43447029750369811759003560764, −6.16179485315492020610358688499, −4.97169851862826787081390191426, −2.66292560046751736284570922934, −1.06914708103473513745467390536,
1.06914708103473513745467390536, 2.66292560046751736284570922934, 4.97169851862826787081390191426, 6.16179485315492020610358688499, 7.43447029750369811759003560764, 8.163877458247526013046625358900, 8.885749534287500360974957337618, 10.03384997744713630229379197811, 10.83919911327928114927134262375, 11.52467847100609427449141012181