| L(s) = 1 | − 0.878·3-s − 5-s + 0.121·7-s − 2.22·9-s − 2.87·11-s − 5.22·13-s + 0.878·15-s − 2.22·17-s − 1.22·19-s − 0.106·21-s + 23-s + 25-s + 4.59·27-s − 9.34·29-s − 2.12·31-s + 2.52·33-s − 0.121·35-s − 5.59·37-s + 4.59·39-s + 8.22·41-s − 8·43-s + 2.22·45-s − 10.4·47-s − 6.98·49-s + 1.95·51-s + 3.59·53-s + 2.87·55-s + ⋯ |
| L(s) = 1 | − 0.507·3-s − 0.447·5-s + 0.0459·7-s − 0.742·9-s − 0.867·11-s − 1.45·13-s + 0.226·15-s − 0.540·17-s − 0.281·19-s − 0.0232·21-s + 0.208·23-s + 0.200·25-s + 0.883·27-s − 1.73·29-s − 0.381·31-s + 0.440·33-s − 0.0205·35-s − 0.919·37-s + 0.735·39-s + 1.28·41-s − 1.21·43-s + 0.332·45-s − 1.52·47-s − 0.997·49-s + 0.274·51-s + 0.493·53-s + 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2480741460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2480741460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.878T + 3T^{2} \) |
| 7 | \( 1 - 0.121T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 5.22T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 8.22T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.59T + 53T^{2} \) |
| 59 | \( 1 - 0.650T + 59T^{2} \) |
| 61 | \( 1 - 7.33T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 3.51T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 0.486T + 89T^{2} \) |
| 97 | \( 1 - 0.635T + 97T^{2} \) |
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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
−7.87839357551744597895469369256, −7.20434495014072268513815200533, −6.59316378100958398031221180865, −5.62314880841538109544244754140, −5.15360301967191961370338007504, −4.53075264838286504597874083313, −3.49781598218975044491537359042, −2.70119155984550668247968726840, −1.88949652003236875534967879854, −0.24000591451875777903474357777,
0.24000591451875777903474357777, 1.88949652003236875534967879854, 2.70119155984550668247968726840, 3.49781598218975044491537359042, 4.53075264838286504597874083313, 5.15360301967191961370338007504, 5.62314880841538109544244754140, 6.59316378100958398031221180865, 7.20434495014072268513815200533, 7.87839357551744597895469369256
