| L(s) = 1 | + 2-s − 0.681·3-s + 4-s − 0.681·6-s + 0.936·7-s + 8-s − 2.53·9-s + 0.0929·11-s − 0.681·12-s − 1.69·13-s + 0.936·14-s + 16-s + 4.42·17-s − 2.53·18-s + 5.52·19-s − 0.638·21-s + 0.0929·22-s + 5.73·23-s − 0.681·24-s − 1.69·26-s + 3.77·27-s + 0.936·28-s + 29-s + 0.970·31-s + 32-s − 0.0633·33-s + 4.42·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.393·3-s + 0.5·4-s − 0.278·6-s + 0.353·7-s + 0.353·8-s − 0.844·9-s + 0.0280·11-s − 0.196·12-s − 0.470·13-s + 0.250·14-s + 0.250·16-s + 1.07·17-s − 0.597·18-s + 1.26·19-s − 0.139·21-s + 0.0198·22-s + 1.19·23-s − 0.139·24-s − 0.332·26-s + 0.726·27-s + 0.176·28-s + 0.185·29-s + 0.174·31-s + 0.176·32-s − 0.0110·33-s + 0.759·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369550762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369550762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.681T + 3T^{2} \) |
| 7 | \( 1 - 0.936T + 7T^{2} \) |
| 11 | \( 1 - 0.0929T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 31 | \( 1 - 0.970T + 31T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.950T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 + 2.36T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 + 7.64T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 9.18T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 5.18T + 97T^{2} \) |
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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
−9.605586337007981313724914343179, −8.680571319308551702312063252116, −7.70798639151942695503482693978, −7.07341429804556016477594038868, −5.95780595329755241801087297174, −5.37083949295003577541516099814, −4.67533333308827362501549838041, −3.40743459578996279142109896322, −2.63902677788731594630958413504, −1.06972222138763499328748791956,
1.06972222138763499328748791956, 2.63902677788731594630958413504, 3.40743459578996279142109896322, 4.67533333308827362501549838041, 5.37083949295003577541516099814, 5.95780595329755241801087297174, 7.07341429804556016477594038868, 7.70798639151942695503482693978, 8.680571319308551702312063252116, 9.605586337007981313724914343179
