| L(s) = 1 | − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s + 4·19-s − 25-s − 2·29-s − 2·35-s − 6·37-s − 2·41-s − 4·43-s + 49-s + 6·53-s + 8·55-s − 12·59-s + 2·61-s − 4·65-s + 4·67-s − 6·73-s − 4·77-s + 16·79-s + 12·83-s − 12·85-s + 14·89-s + 2·91-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.702·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s − 1.30·85-s + 1.48·89-s + 0.209·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473336744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473336744\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.229992153219430708166678625388, −7.70691183501011689426667743544, −7.32441936749744810989380005535, −6.14521652142582152085553341069, −5.34641745697560833577710874285, −4.80722452268725035282926398957, −3.62256199043071801163653681899, −3.22746506277116131252491518684, −1.93480858488759200478154145238, −0.69577924383102779401083388688,
0.69577924383102779401083388688, 1.93480858488759200478154145238, 3.22746506277116131252491518684, 3.62256199043071801163653681899, 4.80722452268725035282926398957, 5.34641745697560833577710874285, 6.14521652142582152085553341069, 7.32441936749744810989380005535, 7.70691183501011689426667743544, 8.229992153219430708166678625388
