| L(s) = 1 | + 3.29·3-s + 3.10·5-s + 7.84·9-s + 3.57·11-s − 2.94·13-s + 10.2·15-s + 0.836·17-s + 1.07·19-s − 4.90·23-s + 4.62·25-s + 15.9·27-s − 3.82·29-s + 5.16·31-s + 11.7·33-s − 6.09·37-s − 9.68·39-s − 41-s + 7.59·43-s + 24.3·45-s − 13.2·47-s + 2.75·51-s − 3.67·53-s + 11.0·55-s + 3.54·57-s + 8.05·59-s + 6.21·61-s − 9.12·65-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 1.38·5-s + 2.61·9-s + 1.07·11-s − 0.815·13-s + 2.63·15-s + 0.202·17-s + 0.246·19-s − 1.02·23-s + 0.924·25-s + 3.07·27-s − 0.709·29-s + 0.927·31-s + 2.05·33-s − 1.00·37-s − 1.55·39-s − 0.156·41-s + 1.15·43-s + 3.62·45-s − 1.93·47-s + 0.385·51-s − 0.504·53-s + 1.49·55-s + 0.468·57-s + 1.04·59-s + 0.796·61-s − 1.13·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.335472966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.335472966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 11 | \( 1 - 3.57T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 - 0.836T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 3.67T + 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 - 6.21T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 8.78T + 73T^{2} \) |
| 79 | \( 1 + 4.46T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.174862406537746063716526883736, −7.05825685601163559488331389696, −6.75092391492652711452811838371, −5.79351678035557013181568321133, −4.92393032276177498216879102240, −4.04529736826693046331793954508, −3.41435900614222095851375923259, −2.50177698677605173476335927867, −1.99874885146491878591735587718, −1.28169501407571667202204788676,
1.28169501407571667202204788676, 1.99874885146491878591735587718, 2.50177698677605173476335927867, 3.41435900614222095851375923259, 4.04529736826693046331793954508, 4.92393032276177498216879102240, 5.79351678035557013181568321133, 6.75092391492652711452811838371, 7.05825685601163559488331389696, 8.174862406537746063716526883736
