| L(s) = 1 | − 2.23·3-s − 2.82·7-s + 2.00·9-s − 2.23·11-s + 6.32·13-s + 5·17-s − 2.23·19-s + 6.32·21-s + 5.65·23-s + 2.23·27-s + 6.32·29-s + 5.00·33-s − 6.32·37-s − 14.1·39-s − 3·41-s − 8.94·43-s + 2.82·47-s + 1.00·49-s − 11.1·51-s − 12.6·53-s + 5.00·57-s − 8.94·59-s − 6.32·61-s − 5.65·63-s + 11.1·67-s − 12.6·69-s − 14.1·71-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 1.06·7-s + 0.666·9-s − 0.674·11-s + 1.75·13-s + 1.21·17-s − 0.512·19-s + 1.38·21-s + 1.17·23-s + 0.430·27-s + 1.17·29-s + 0.870·33-s − 1.03·37-s − 2.26·39-s − 0.468·41-s − 1.36·43-s + 0.412·47-s + 0.142·49-s − 1.56·51-s − 1.73·53-s + 0.662·57-s − 1.16·59-s − 0.809·61-s − 0.712·63-s + 1.36·67-s − 1.52·69-s − 1.67·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9003329730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9003329730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 15T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.092775928112066617671512937075, −7.00311004868073076408865416301, −6.44434736292504726906825067357, −6.01410478878777487362416035358, −5.28255496174374061458410899699, −4.65243363819386399413904217610, −3.42619973582313701232089350359, −3.08859854051818452993117504779, −1.49612872681251295947700160826, −0.55563200358841687574981577083,
0.55563200358841687574981577083, 1.49612872681251295947700160826, 3.08859854051818452993117504779, 3.42619973582313701232089350359, 4.65243363819386399413904217610, 5.28255496174374061458410899699, 6.01410478878777487362416035358, 6.44434736292504726906825067357, 7.00311004868073076408865416301, 8.092775928112066617671512937075
