| L(s) = 1 | + 1.34·2-s − 0.184·4-s − 2.53·5-s − 0.184·7-s − 2.94·8-s − 3.41·10-s + 3.57·11-s − 6.53·13-s − 0.248·14-s − 3.59·16-s + 4.63·19-s + 0.467·20-s + 4.81·22-s − 3.10·23-s + 1.41·25-s − 8.80·26-s + 0.0341·28-s + 6.35·29-s + 7.41·31-s + 1.04·32-s + 0.467·35-s + 8.39·37-s + 6.24·38-s + 7.45·40-s + 7.18·41-s − 4.93·43-s − 0.660·44-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0923·4-s − 1.13·5-s − 0.0698·7-s − 1.04·8-s − 1.07·10-s + 1.07·11-s − 1.81·13-s − 0.0665·14-s − 0.899·16-s + 1.06·19-s + 0.104·20-s + 1.02·22-s − 0.647·23-s + 0.282·25-s − 1.72·26-s + 0.00645·28-s + 1.18·29-s + 1.33·31-s + 0.184·32-s + 0.0790·35-s + 1.38·37-s + 1.01·38-s + 1.17·40-s + 1.12·41-s − 0.752·43-s − 0.0995·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696542274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696542274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 + 0.184T + 7T^{2} \) |
| 11 | \( 1 - 3.57T + 11T^{2} \) |
| 13 | \( 1 + 6.53T + 13T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 8.39T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 + 8.41T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 + 1.32T + 89T^{2} \) |
| 97 | \( 1 - 8.75T + 97T^{2} \) |
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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
−8.888380785215105250766034618026, −7.960037567743194732497630384359, −7.34426273904554500635909333577, −6.48343170367786851403031707131, −5.64521581809653549746328122178, −4.50434642211270093131585686731, −4.41559881633446680023720761219, −3.34056862667916498576503730948, −2.57127143132137964731348799635, −0.69935941798842322211311813248,
0.69935941798842322211311813248, 2.57127143132137964731348799635, 3.34056862667916498576503730948, 4.41559881633446680023720761219, 4.50434642211270093131585686731, 5.64521581809653549746328122178, 6.48343170367786851403031707131, 7.34426273904554500635909333577, 7.960037567743194732497630384359, 8.888380785215105250766034618026
