| L(s) = 1 | − 2.57·5-s + 7-s + 1.35·11-s − 4.19·13-s − 4.92·17-s + 19-s − 6.19·23-s + 1.62·25-s − 0.760·29-s + 9.46·31-s − 2.57·35-s − 5.27·37-s − 1.23·41-s − 9.91·43-s + 5.77·47-s + 49-s + 1.15·53-s − 3.48·55-s + 7.88·59-s − 8.31·61-s + 10.8·65-s + 15.3·67-s − 6.56·71-s − 9.02·73-s + 1.35·77-s − 0.707·79-s + 11.6·83-s + ⋯ |
| L(s) = 1 | − 1.15·5-s + 0.377·7-s + 0.407·11-s − 1.16·13-s − 1.19·17-s + 0.229·19-s − 1.29·23-s + 0.324·25-s − 0.141·29-s + 1.70·31-s − 0.435·35-s − 0.867·37-s − 0.193·41-s − 1.51·43-s + 0.841·47-s + 0.142·49-s + 0.158·53-s − 0.469·55-s + 1.02·59-s − 1.06·61-s + 1.33·65-s + 1.87·67-s − 0.778·71-s − 1.05·73-s + 0.154·77-s − 0.0795·79-s + 1.27·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9125806840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9125806840\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 + 0.760T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 + 9.91T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 1.15T + 53T^{2} \) |
| 59 | \( 1 - 7.88T + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 + 9.02T + 73T^{2} \) |
| 79 | \( 1 + 0.707T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.74374090962076976375472826303, −7.02722837331985164333216579958, −6.54327037785055120154960960364, −5.56119014430158092007064185613, −4.70581941356576499022890178902, −4.29047382975501324246701224055, −3.55968554876198332035565650594, −2.59993878955629666568487667013, −1.78355952457246901815361177783, −0.44193107534721100443126647794,
0.44193107534721100443126647794, 1.78355952457246901815361177783, 2.59993878955629666568487667013, 3.55968554876198332035565650594, 4.29047382975501324246701224055, 4.70581941356576499022890178902, 5.56119014430158092007064185613, 6.54327037785055120154960960364, 7.02722837331985164333216579958, 7.74374090962076976375472826303
