| L(s) = 1 | + 3.06·3-s + 5-s − 7-s + 6.41·9-s + 4·11-s − 13-s + 3.06·15-s + 0.720·17-s + 1.27·19-s − 3.06·21-s − 2·23-s + 25-s + 10.4·27-s − 3.06·29-s − 5.76·31-s + 12.2·33-s − 35-s + 9.20·37-s − 3.06·39-s + 3.27·41-s + 10.1·43-s + 6.41·45-s + 0.696·47-s + 49-s + 2.21·51-s − 4.69·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 0.447·5-s − 0.377·7-s + 2.13·9-s + 1.20·11-s − 0.277·13-s + 0.792·15-s + 0.174·17-s + 0.293·19-s − 0.669·21-s − 0.417·23-s + 0.200·25-s + 2.01·27-s − 0.569·29-s − 1.03·31-s + 2.13·33-s − 0.169·35-s + 1.51·37-s − 0.491·39-s + 0.512·41-s + 1.54·43-s + 0.956·45-s + 0.101·47-s + 0.142·49-s + 0.309·51-s − 0.645·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.288288646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288288646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 0.720T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 3.06T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 - 9.20T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 0.696T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 + 0.559T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 5.41T + 79T^{2} \) |
| 83 | \( 1 + 8.97T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 0.137T + 97T^{2} \) |
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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
−8.739702716578145457449011830261, −7.77172144269216429112247003180, −7.32632176753238582594232658487, −6.46108005306468413174403871080, −5.62578122579056496116568048032, −4.31695829767524646249091564099, −3.81759977572805590092955717782, −2.93316501587958218262146180407, −2.18342485144138676869917942614, −1.23938570573999600085106134619,
1.23938570573999600085106134619, 2.18342485144138676869917942614, 2.93316501587958218262146180407, 3.81759977572805590092955717782, 4.31695829767524646249091564099, 5.62578122579056496116568048032, 6.46108005306468413174403871080, 7.32632176753238582594232658487, 7.77172144269216429112247003180, 8.739702716578145457449011830261
