L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s + 15-s + 6·17-s − 4·19-s − 23-s + 25-s − 27-s + 8·31-s + 2·33-s − 6·37-s − 2·41-s + 2·43-s − 45-s − 4·47-s − 7·49-s − 6·51-s − 2·53-s + 2·55-s + 4·57-s − 2·61-s + 2·67-s + 69-s + 10·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.43·31-s + 0.348·33-s − 0.986·37-s − 0.312·41-s + 0.304·43-s − 0.149·45-s − 0.583·47-s − 49-s − 0.840·51-s − 0.274·53-s + 0.269·55-s + 0.529·57-s − 0.256·61-s + 0.244·67-s + 0.120·69-s + 1.18·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.148626999\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148626999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.048906934798322009234619329638, −7.53500334508667132455642259504, −6.63993182426038670131940243934, −6.05028344747804536628513213157, −5.19326738464827222239357750793, −4.63953943295683838646113776332, −3.69408540630327060152215655702, −2.92471425229465818669339625435, −1.76171685335801410781286989614, −0.59164844342346600835950335745,
0.59164844342346600835950335745, 1.76171685335801410781286989614, 2.92471425229465818669339625435, 3.69408540630327060152215655702, 4.63953943295683838646113776332, 5.19326738464827222239357750793, 6.05028344747804536628513213157, 6.63993182426038670131940243934, 7.53500334508667132455642259504, 8.048906934798322009234619329638