L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 9-s + 10-s + 2·13-s + 16-s + 2·17-s + 18-s − 20-s + 25-s − 2·26-s − 32-s − 2·34-s − 36-s − 37-s + 40-s + 2·41-s + 45-s − 49-s − 50-s + 2·52-s + 64-s − 2·65-s + 2·68-s + 72-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 9-s + 10-s + 2·13-s + 16-s + 2·17-s + 18-s − 20-s + 25-s − 2·26-s − 32-s − 2·34-s − 36-s − 37-s + 40-s + 2·41-s + 45-s − 49-s − 50-s + 2·52-s + 64-s − 2·65-s + 2·68-s + 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5550085472\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5550085472\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )^{2} \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−10.76951103294122934050343824296, −9.627294368993139585609309738254, −8.634989788798236018769708547483, −8.197180060129591003700495988194, −7.42861804446467517653931711123, −6.26623549826840885376822986374, −5.52447345227908919914544570395, −3.73618443577603252210642390799, −3.03517940829376024417291870661, −1.14392418938186635586164799648,
1.14392418938186635586164799648, 3.03517940829376024417291870661, 3.73618443577603252210642390799, 5.52447345227908919914544570395, 6.26623549826840885376822986374, 7.42861804446467517653931711123, 8.197180060129591003700495988194, 8.634989788798236018769708547483, 9.627294368993139585609309738254, 10.76951103294122934050343824296