L(s) = 1 | + 9-s − 13-s − 2·17-s + 25-s − 2·29-s − 49-s + 2·53-s + 2·61-s + 81-s + 2·101-s − 2·113-s − 117-s + ⋯ |
L(s) = 1 | + 9-s − 13-s − 2·17-s + 25-s − 2·29-s − 49-s + 2·53-s + 2·61-s + 81-s + 2·101-s − 2·113-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7040449453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7040449453\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.84216826972720396534944438557, −11.62074117916663017153709902749, −10.69181790496113516813530649640, −9.676660089512915951413231397337, −8.782672525949915380726097618143, −7.40350736653843378830348360900, −6.67707391086418863494569206827, −5.10538667487405721881327138549, −4.04535655824561171760116937879, −2.20467624493930931798195661261,
2.20467624493930931798195661261, 4.04535655824561171760116937879, 5.10538667487405721881327138549, 6.67707391086418863494569206827, 7.40350736653843378830348360900, 8.782672525949915380726097618143, 9.676660089512915951413231397337, 10.69181790496113516813530649640, 11.62074117916663017153709902749, 12.84216826972720396534944438557