| L(s) = 1 | − 0.618·3-s + 4.85·7-s − 2.61·9-s + 3.38·11-s − 0.381·13-s + 5.85·17-s + 6.85·19-s − 3.00·21-s + 23-s + 3.47·27-s + 3.70·29-s + 8.85·31-s − 2.09·33-s − 3.70·37-s + 0.236·39-s − 3.38·41-s + 6.76·43-s − 11.7·47-s + 16.5·49-s − 3.61·51-s + 2·53-s − 4.23·57-s + 6·59-s − 3.85·61-s − 12.7·63-s + 0.763·67-s − 0.618·69-s + ⋯ |
| L(s) = 1 | − 0.356·3-s + 1.83·7-s − 0.872·9-s + 1.01·11-s − 0.105·13-s + 1.41·17-s + 1.57·19-s − 0.654·21-s + 0.208·23-s + 0.668·27-s + 0.688·29-s + 1.59·31-s − 0.363·33-s − 0.609·37-s + 0.0378·39-s − 0.528·41-s + 1.03·43-s − 1.70·47-s + 2.36·49-s − 0.506·51-s + 0.274·53-s − 0.561·57-s + 0.781·59-s − 0.493·61-s − 1.60·63-s + 0.0933·67-s − 0.0744·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.872769609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.872769609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 0.381T + 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 + 9.70T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.86865038545985921417933507348, −7.09325830199187544717824784188, −6.29187798516764691942004930954, −5.43620966039156213947596491617, −5.13314637995467359070377944920, −4.38458250908403905248070270349, −3.42735965729922102484923547998, −2.65353596535418697113938178495, −1.42504954846199028369721397294, −0.978525428787221540032160992044,
0.978525428787221540032160992044, 1.42504954846199028369721397294, 2.65353596535418697113938178495, 3.42735965729922102484923547998, 4.38458250908403905248070270349, 5.13314637995467359070377944920, 5.43620966039156213947596491617, 6.29187798516764691942004930954, 7.09325830199187544717824784188, 7.86865038545985921417933507348
