| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 5·11-s − 4·13-s − 15-s − 3·17-s − 19-s + 21-s − 4·25-s − 27-s + 4·29-s − 2·31-s − 5·33-s − 35-s − 4·37-s + 4·39-s − 4·41-s + 5·43-s + 45-s − 7·47-s − 6·49-s + 3·51-s + 2·53-s + 5·55-s + 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 1.10·13-s − 0.258·15-s − 0.727·17-s − 0.229·19-s + 0.218·21-s − 4/5·25-s − 0.192·27-s + 0.742·29-s − 0.359·31-s − 0.870·33-s − 0.169·35-s − 0.657·37-s + 0.640·39-s − 0.624·41-s + 0.762·43-s + 0.149·45-s − 1.02·47-s − 6/7·49-s + 0.420·51-s + 0.274·53-s + 0.674·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.207763466322965873716467162773, −7.15344641458308162055100896622, −6.64158471640639013042779348027, −6.06507279155339670095112576122, −5.14971454545867607032763212297, −4.39480131765469361117634281051, −3.57509689533885776423548420295, −2.37697055012752399564373659845, −1.43670953857849266104113550140, 0,
1.43670953857849266104113550140, 2.37697055012752399564373659845, 3.57509689533885776423548420295, 4.39480131765469361117634281051, 5.14971454545867607032763212297, 6.06507279155339670095112576122, 6.64158471640639013042779348027, 7.15344641458308162055100896622, 8.207763466322965873716467162773
