| L(s) = 1 | + 1.30·5-s + 7-s − 5.88·11-s + 3.78·13-s − 5.60·17-s + 4.40·19-s − 23-s − 3.30·25-s − 0.202·29-s − 0.725·31-s + 1.30·35-s + 1.52·37-s − 6.47·41-s + 1.44·43-s + 49-s + 1.66·53-s − 7.66·55-s + 2.53·59-s − 5.18·61-s + 4.92·65-s − 5.57·67-s − 10.5·71-s − 11.6·73-s − 5.88·77-s + 8.23·79-s − 1.34·83-s − 7.30·85-s + ⋯ |
| L(s) = 1 | + 0.582·5-s + 0.377·7-s − 1.77·11-s + 1.04·13-s − 1.35·17-s + 1.01·19-s − 0.208·23-s − 0.660·25-s − 0.0376·29-s − 0.130·31-s + 0.220·35-s + 0.250·37-s − 1.01·41-s + 0.221·43-s + 0.142·49-s + 0.228·53-s − 1.03·55-s + 0.329·59-s − 0.663·61-s + 0.610·65-s − 0.681·67-s − 1.25·71-s − 1.36·73-s − 0.670·77-s + 0.926·79-s − 0.148·83-s − 0.792·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 + 5.88T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 29 | \( 1 + 0.202T + 29T^{2} \) |
| 31 | \( 1 + 0.725T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 1.44T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.66T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 + 1.34T + 83T^{2} \) |
| 89 | \( 1 + 9.10T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.77960024791818478362776673027, −7.13872739821921557022576148287, −6.17933375118335584033755985290, −5.61676908927274486068039837939, −4.96713052018036179240957018374, −4.14332470247672091978161396345, −3.09841650937240207264300902605, −2.32919786293090427361248605026, −1.45874322268540723541885343173, 0,
1.45874322268540723541885343173, 2.32919786293090427361248605026, 3.09841650937240207264300902605, 4.14332470247672091978161396345, 4.96713052018036179240957018374, 5.61676908927274486068039837939, 6.17933375118335584033755985290, 7.13872739821921557022576148287, 7.77960024791818478362776673027
