L(s) = 1 | + 5-s − 7-s − 3·11-s + 3·13-s − 3·17-s + 23-s + 25-s + 9·29-s − 2·31-s − 35-s − 4·37-s − 4·41-s + 2·43-s + 13·47-s + 49-s + 2·53-s − 3·55-s − 6·59-s − 6·61-s + 3·65-s + 12·67-s − 8·71-s − 6·73-s + 3·77-s + 13·79-s + 12·83-s − 3·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.832·13-s − 0.727·17-s + 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.359·31-s − 0.169·35-s − 0.657·37-s − 0.624·41-s + 0.304·43-s + 1.89·47-s + 1/7·49-s + 0.274·53-s − 0.404·55-s − 0.781·59-s − 0.768·61-s + 0.372·65-s + 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.341·77-s + 1.46·79-s + 1.31·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.179282197\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.179282197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.46184764445539, −13.27573532798922, −12.73268830141195, −12.04210190269012, −11.90202970358865, −10.86715627299210, −10.67945613258287, −10.40856883313414, −9.634143237426775, −9.188115169483613, −8.690876753655632, −8.263567199720459, −7.663119654829254, −7.039276728219622, −6.549543761895103, −6.113189327512672, −5.505666552664774, −5.030175620909865, −4.411834709883488, −3.803384174745414, −3.113353457681651, −2.606708149776291, −2.022858768408325, −1.209353940099876, −0.4737205530373174,
0.4737205530373174, 1.209353940099876, 2.022858768408325, 2.606708149776291, 3.113353457681651, 3.803384174745414, 4.411834709883488, 5.030175620909865, 5.505666552664774, 6.113189327512672, 6.549543761895103, 7.039276728219622, 7.663119654829254, 8.263567199720459, 8.690876753655632, 9.188115169483613, 9.634143237426775, 10.40856883313414, 10.67945613258287, 10.86715627299210, 11.90202970358865, 12.04210190269012, 12.73268830141195, 13.27573532798922, 13.46184764445539