| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 3·11-s − 4·13-s + 8·17-s − 19-s − 8·21-s + 4·27-s − 2·29-s + 5·31-s + 6·33-s + 2·37-s + 8·39-s + 5·41-s + 4·43-s + 2·47-s + 9·49-s − 16·51-s − 8·53-s + 2·57-s + 12·59-s + 3·61-s + 4·63-s − 2·67-s + 5·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 1.94·17-s − 0.229·19-s − 1.74·21-s + 0.769·27-s − 0.371·29-s + 0.898·31-s + 1.04·33-s + 0.328·37-s + 1.28·39-s + 0.780·41-s + 0.609·43-s + 0.291·47-s + 9/7·49-s − 2.24·51-s − 1.09·53-s + 0.264·57-s + 1.56·59-s + 0.384·61-s + 0.503·63-s − 0.244·67-s + 0.593·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.919923316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919923316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.75322706061234, −12.36928571676920, −12.09057493773901, −11.59786562494991, −11.15543693140908, −10.78147250016880, −10.33550319868552, −9.831028342860828, −9.464689094983534, −8.563186885084525, −8.145781372268282, −7.751220071732520, −7.402796712623142, −6.797054958717748, −6.055720998476561, −5.637153815402688, −5.164191183190584, −4.983550914914529, −4.387462498567642, −3.757634954738395, −2.830354791607402, −2.487303119294370, −1.687461112689293, −1.001467580399543, −0.4963303751628695,
0.4963303751628695, 1.001467580399543, 1.687461112689293, 2.487303119294370, 2.830354791607402, 3.757634954738395, 4.387462498567642, 4.983550914914529, 5.164191183190584, 5.637153815402688, 6.055720998476561, 6.797054958717748, 7.402796712623142, 7.751220071732520, 8.145781372268282, 8.563186885084525, 9.464689094983534, 9.831028342860828, 10.33550319868552, 10.78147250016880, 11.15543693140908, 11.59786562494991, 12.09057493773901, 12.36928571676920, 12.75322706061234
