| L(s) = 1 | − 2·4-s − 3·5-s + 7-s + 11-s − 4·13-s + 4·16-s + 6·17-s + 2·19-s + 6·20-s − 3·23-s + 4·25-s − 2·28-s + 6·29-s + 5·31-s − 3·35-s + 11·37-s − 6·41-s + 8·43-s − 2·44-s + 49-s + 8·52-s + 6·53-s − 3·55-s + 9·59-s − 10·61-s − 8·64-s + 12·65-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s + 0.377·7-s + 0.301·11-s − 1.10·13-s + 16-s + 1.45·17-s + 0.458·19-s + 1.34·20-s − 0.625·23-s + 4/5·25-s − 0.377·28-s + 1.11·29-s + 0.898·31-s − 0.507·35-s + 1.80·37-s − 0.937·41-s + 1.21·43-s − 0.301·44-s + 1/7·49-s + 1.10·52-s + 0.824·53-s − 0.404·55-s + 1.17·59-s − 1.28·61-s − 64-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8965723977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8965723977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−10.24457963817336821430305627863, −9.694504921717345364920145695912, −8.556667066253414749137472853294, −7.901671452709333870689471023015, −7.32024742481712425147757589138, −5.80492978659389909733109582978, −4.73710013360513636713787424546, −4.10975481209818427421549838760, −3.01251424255196705097668786615, −0.821355075233086309329735836862,
0.821355075233086309329735836862, 3.01251424255196705097668786615, 4.10975481209818427421549838760, 4.73710013360513636713787424546, 5.80492978659389909733109582978, 7.32024742481712425147757589138, 7.901671452709333870689471023015, 8.556667066253414749137472853294, 9.694504921717345364920145695912, 10.24457963817336821430305627863
