| L(s) = 1 | − 1.18·2-s − 3-s − 0.592·4-s − 5-s + 1.18·6-s + 4.49·7-s + 3.07·8-s + 9-s + 1.18·10-s − 6.47·11-s + 0.592·12-s − 2.92·13-s − 5.33·14-s + 15-s − 2.46·16-s − 1.67·17-s − 1.18·18-s + 0.592·20-s − 4.49·21-s + 7.67·22-s − 8.57·23-s − 3.07·24-s + 25-s + 3.46·26-s − 27-s − 2.66·28-s − 8.22·29-s + ⋯ |
| L(s) = 1 | − 0.838·2-s − 0.577·3-s − 0.296·4-s − 0.447·5-s + 0.484·6-s + 1.69·7-s + 1.08·8-s + 0.333·9-s + 0.375·10-s − 1.95·11-s + 0.171·12-s − 0.810·13-s − 1.42·14-s + 0.258·15-s − 0.615·16-s − 0.405·17-s − 0.279·18-s + 0.132·20-s − 0.981·21-s + 1.63·22-s − 1.78·23-s − 0.627·24-s + 0.200·25-s + 0.679·26-s − 0.192·27-s − 0.503·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4026136714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4026136714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 7 | \( 1 - 4.49T + 7T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 + 8.22T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 37 | \( 1 - 0.0985T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 - 0.922T + 59T^{2} \) |
| 61 | \( 1 + 6.35T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 5.04T + 97T^{2} \) |
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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
−7.996959677735951580475489990499, −7.78053706151529661061800818399, −7.18504221982360790080849199420, −5.83192035208270537086127395215, −5.11879064273052265401971196578, −4.70060328916272448656058798953, −3.99540614941716856671999565797, −2.42878133269364813577948992968, −1.72069822819598004833306838819, −0.39725981633737071657272328007,
0.39725981633737071657272328007, 1.72069822819598004833306838819, 2.42878133269364813577948992968, 3.99540614941716856671999565797, 4.70060328916272448656058798953, 5.11879064273052265401971196578, 5.83192035208270537086127395215, 7.18504221982360790080849199420, 7.78053706151529661061800818399, 7.996959677735951580475489990499
