| L(s) = 1 | − 3·3-s − 2·5-s + 6·9-s + 5·11-s + 7·13-s + 6·15-s − 17-s + 2·19-s − 25-s − 9·27-s − 6·29-s + 4·31-s − 15·33-s + 8·37-s − 21·39-s + 2·41-s + 8·43-s − 12·45-s − 10·47-s + 3·51-s + 3·53-s − 10·55-s − 6·57-s − 12·61-s − 14·65-s + 2·67-s − 71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s + 2·9-s + 1.50·11-s + 1.94·13-s + 1.54·15-s − 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.73·27-s − 1.11·29-s + 0.718·31-s − 2.61·33-s + 1.31·37-s − 3.36·39-s + 0.312·41-s + 1.21·43-s − 1.78·45-s − 1.45·47-s + 0.420·51-s + 0.412·53-s − 1.34·55-s − 0.794·57-s − 1.53·61-s − 1.73·65-s + 0.244·67-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146317487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146317487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.52302527754722, −13.83980860034422, −13.28173239879062, −12.80974630849655, −12.19285541378714, −11.68407657194448, −11.46619402404985, −11.00159868955627, −10.73030889333012, −9.770032551632455, −9.424140621796663, −8.703741667261129, −8.124099181997970, −7.466230780190200, −6.885755664443868, −6.379334569687012, −5.906035185035232, −5.601527727791619, −4.544755224516389, −4.195544060627674, −3.808607094170241, −3.029294025145286, −1.614990594337569, −1.202075266593297, −0.4908803065005152,
0.4908803065005152, 1.202075266593297, 1.614990594337569, 3.029294025145286, 3.808607094170241, 4.195544060627674, 4.544755224516389, 5.601527727791619, 5.906035185035232, 6.379334569687012, 6.885755664443868, 7.466230780190200, 8.124099181997970, 8.703741667261129, 9.424140621796663, 9.770032551632455, 10.73030889333012, 11.00159868955627, 11.46619402404985, 11.68407657194448, 12.19285541378714, 12.80974630849655, 13.28173239879062, 13.83980860034422, 14.52302527754722
