L(s) = 1 | + 3-s + 5-s + 2·7-s − 2·9-s + 3·11-s − 13-s + 15-s + 2·21-s + 4·23-s − 4·25-s − 5·27-s − 29-s + 3·31-s + 3·33-s + 2·35-s − 8·37-s − 39-s − 6·41-s − 5·43-s − 2·45-s + 3·47-s − 3·49-s + 5·53-s + 3·55-s − 8·59-s − 4·63-s − 65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s + 0.436·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s − 0.185·29-s + 0.538·31-s + 0.522·33-s + 0.338·35-s − 1.31·37-s − 0.160·39-s − 0.937·41-s − 0.762·43-s − 0.298·45-s + 0.437·47-s − 3/7·49-s + 0.686·53-s + 0.404·55-s − 1.04·59-s − 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.594638024\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.594638024\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.03225617833127401634294644320, −11.35936445667626245776837847935, −10.17646720474260025332202942321, −9.109468260421844793465351652049, −8.430240929203566777866130945662, −7.28385569168970544975317725065, −6.02226938483447033650045844097, −4.83506868338445543793010149058, −3.35801131767532934555280418053, −1.86240791565196501569145297442,
1.86240791565196501569145297442, 3.35801131767532934555280418053, 4.83506868338445543793010149058, 6.02226938483447033650045844097, 7.28385569168970544975317725065, 8.430240929203566777866130945662, 9.109468260421844793465351652049, 10.17646720474260025332202942321, 11.35936445667626245776837847935, 12.03225617833127401634294644320