| L(s) = 1 | + 1.72·3-s + 0.138·5-s − 0.00725·9-s + 2.69·11-s + 5.05·13-s + 0.239·15-s + 4.38·17-s − 5.73·19-s + 8.09·23-s − 4.98·25-s − 5.20·27-s + 3.49·29-s + 2.84·31-s + 4.65·33-s − 4.04·37-s + 8.73·39-s + 41-s − 10.8·43-s − 0.00100·45-s − 0.549·47-s + 7.58·51-s + 11.8·53-s + 0.372·55-s − 9.91·57-s + 9.85·59-s + 7.09·61-s + 0.698·65-s + ⋯ |
| L(s) = 1 | + 0.998·3-s + 0.0618·5-s − 0.00241·9-s + 0.812·11-s + 1.40·13-s + 0.0617·15-s + 1.06·17-s − 1.31·19-s + 1.68·23-s − 0.996·25-s − 1.00·27-s + 0.648·29-s + 0.511·31-s + 0.811·33-s − 0.665·37-s + 1.39·39-s + 0.156·41-s − 1.65·43-s − 0.000149·45-s − 0.0801·47-s + 1.06·51-s + 1.63·53-s + 0.0502·55-s − 1.31·57-s + 1.28·59-s + 0.908·61-s + 0.0866·65-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
| \(L(1)\) |
\(\approx\) |
\(3.494308242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.494308242\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 0.138T + 5T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 - 8.09T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.549T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 9.85T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 7.81T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
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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
−8.151827594122396358102171169032, −7.10948855146574501898304683328, −6.54506324689659066376713154842, −5.81817976844318104677271839459, −5.03560972686145046602791736508, −3.89493753788066566140847887959, −3.61277779689411319468722405204, −2.74520742400250981971648887924, −1.83282893929898172799115342578, −0.923406208386764410315545996406,
0.923406208386764410315545996406, 1.83282893929898172799115342578, 2.74520742400250981971648887924, 3.61277779689411319468722405204, 3.89493753788066566140847887959, 5.03560972686145046602791736508, 5.81817976844318104677271839459, 6.54506324689659066376713154842, 7.10948855146574501898304683328, 8.151827594122396358102171169032
