| L(s) = 1 | − 2·5-s + 11-s + 4·13-s + 6·17-s + 2·19-s − 25-s − 6·29-s − 2·31-s − 2·37-s − 2·41-s + 12·43-s − 6·47-s + 10·53-s − 2·55-s − 8·59-s + 4·61-s − 8·65-s + 8·71-s − 6·73-s + 12·79-s − 6·83-s − 12·85-s − 12·89-s − 4·95-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.458·19-s − 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.328·37-s − 0.312·41-s + 1.82·43-s − 0.875·47-s + 1.37·53-s − 0.269·55-s − 1.04·59-s + 0.512·61-s − 0.992·65-s + 0.949·71-s − 0.702·73-s + 1.35·79-s − 0.658·83-s − 1.30·85-s − 1.27·89-s − 0.410·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.060579078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060579078\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.63244219208787, −12.09286782049540, −11.78225550057447, −11.32637483939278, −10.89485343765855, −10.49154285952626, −9.886730909061209, −9.332505965620565, −9.111861626699573, −8.262845926028218, −8.142541434677356, −7.553741865690251, −7.195549598780497, −6.639411124632797, −5.989654030342663, −5.542073692876122, −5.253159813844713, −4.326268211380972, −3.964076503342889, −3.549925171786955, −3.136491131533470, −2.394258685804852, −1.565400580932576, −1.136800899922508, −0.4184515019693010,
0.4184515019693010, 1.136800899922508, 1.565400580932576, 2.394258685804852, 3.136491131533470, 3.549925171786955, 3.964076503342889, 4.326268211380972, 5.253159813844713, 5.542073692876122, 5.989654030342663, 6.639411124632797, 7.195549598780497, 7.553741865690251, 8.142541434677356, 8.262845926028218, 9.111861626699573, 9.332505965620565, 9.886730909061209, 10.49154285952626, 10.89485343765855, 11.32637483939278, 11.78225550057447, 12.09286782049540, 12.63244219208787
