| L(s) = 1 | − 5-s + 3.37·7-s + 1.37·11-s + 2·13-s + 17-s + 3.37·19-s + 2.74·23-s + 25-s − 1.37·29-s + 2·31-s − 3.37·35-s − 8.11·37-s − 1.37·41-s − 0.744·43-s + 1.37·47-s + 4.37·49-s − 1.37·53-s − 1.37·55-s + 2·61-s − 2·65-s + 4.74·67-s + 8.74·71-s + 12.1·73-s + 4.62·77-s + 2·79-s − 8.74·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.27·7-s + 0.413·11-s + 0.554·13-s + 0.242·17-s + 0.773·19-s + 0.572·23-s + 0.200·25-s − 0.254·29-s + 0.359·31-s − 0.570·35-s − 1.33·37-s − 0.214·41-s − 0.113·43-s + 0.200·47-s + 0.624·49-s − 0.188·53-s − 0.185·55-s + 0.256·61-s − 0.248·65-s + 0.579·67-s + 1.03·71-s + 1.41·73-s + 0.527·77-s + 0.225·79-s − 0.959·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.205940537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.205940537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 + 0.744T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 - 8.74T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.568301564559790888091049261284, −8.038141049834676860438271498125, −7.29995065947815300889848273884, −6.55913286945837621252001546086, −5.46564203922724869278780694628, −4.91345411169326549668973905911, −3.99185750978006879703160537403, −3.20667147255068633574649695099, −1.90526923566986965213755877380, −0.970128666782423077784489691372,
0.970128666782423077784489691372, 1.90526923566986965213755877380, 3.20667147255068633574649695099, 3.99185750978006879703160537403, 4.91345411169326549668973905911, 5.46564203922724869278780694628, 6.55913286945837621252001546086, 7.29995065947815300889848273884, 8.038141049834676860438271498125, 8.568301564559790888091049261284
