L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.5 − 1.53i)3-s + (0.309 − 0.951i)4-s + (1.30 + 0.951i)6-s + (0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (0.309 + 0.951i)11-s − 1.61·12-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s + (0.5 − 1.53i)18-s + (0.190 + 0.587i)19-s + (−0.809 − 0.587i)22-s + (1.30 − 0.951i)24-s + (0.309 + 0.951i)25-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.5 − 1.53i)3-s + (0.309 − 0.951i)4-s + (1.30 + 0.951i)6-s + (0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (0.309 + 0.951i)11-s − 1.61·12-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s + (0.5 − 1.53i)18-s + (0.190 + 0.587i)19-s + (−0.809 − 0.587i)22-s + (1.30 − 0.951i)24-s + (0.309 + 0.951i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3471379073\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3471379073\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
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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
−14.31913077809594559769950473543, −13.24841791962501967341869657467, −12.13455463741364210143205941073, −11.20692116359456204291987773293, −9.801542251643198264879174078581, −8.433898624247411372751716098663, −7.28039020908791378434224755870, −6.66545754783887735413245583256, −5.32559889154914950171752856996, −1.80284319275365030447112589300,
3.29597448370317997586963941688, 4.63376475369904117861461445186, 6.38456810366459715900409676471, 8.341090551312655104382721172892, 9.284969942379507859608611312768, 10.25751161800844311628585574119, 11.07767912405533495553158013088, 11.82681198195979590832263812253, 13.34850789296762653892449518238, 14.86309983140617749909332123927