L(s) = 1 | + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (0.856 + 0.515i)5-s + (−0.0541 + 0.998i)7-s + (0.856 − 0.515i)12-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.108 + 1.99i)17-s + (0.725 − 0.687i)19-s + (0.947 − 0.319i)20-s + (−0.370 + 0.928i)21-s + (−0.561 − 0.827i)27-s + (0.725 + 0.687i)28-s + (−0.907 − 0.419i)29-s + (−0.561 + 0.827i)35-s + ⋯ |
L(s) = 1 | + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (0.856 + 0.515i)5-s + (−0.0541 + 0.998i)7-s + (0.856 − 0.515i)12-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.108 + 1.99i)17-s + (0.725 − 0.687i)19-s + (0.947 − 0.319i)20-s + (−0.370 + 0.928i)21-s + (−0.561 − 0.827i)27-s + (0.725 + 0.687i)28-s + (−0.907 − 0.419i)29-s + (−0.561 + 0.827i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.294918038\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.294918038\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.647 + 0.762i)T^{2} \) |
| 3 | \( 1 + (-0.947 - 0.319i)T + (0.796 + 0.605i)T^{2} \) |
| 5 | \( 1 + (-0.856 - 0.515i)T + (0.468 + 0.883i)T^{2} \) |
| 7 | \( 1 + (0.0541 - 0.998i)T + (-0.994 - 0.108i)T^{2} \) |
| 11 | \( 1 + (0.947 + 0.319i)T^{2} \) |
| 13 | \( 1 + (-0.267 + 0.963i)T^{2} \) |
| 17 | \( 1 + (-0.108 - 1.99i)T + (-0.994 + 0.108i)T^{2} \) |
| 19 | \( 1 + (-0.725 + 0.687i)T + (0.0541 - 0.998i)T^{2} \) |
| 23 | \( 1 + (-0.907 + 0.419i)T^{2} \) |
| 29 | \( 1 + (0.907 + 0.419i)T + (0.647 + 0.762i)T^{2} \) |
| 31 | \( 1 + (-0.0541 - 0.998i)T^{2} \) |
| 37 | \( 1 + (0.856 - 0.515i)T^{2} \) |
| 41 | \( 1 + (0.976 + 0.214i)T + (0.907 + 0.419i)T^{2} \) |
| 43 | \( 1 + (0.947 - 0.319i)T^{2} \) |
| 47 | \( 1 + (-0.468 + 0.883i)T^{2} \) |
| 53 | \( 1 + (-0.994 + 0.108i)T + (0.976 - 0.214i)T^{2} \) |
| 61 | \( 1 + (-0.647 + 0.762i)T^{2} \) |
| 67 | \( 1 + (0.856 + 0.515i)T^{2} \) |
| 71 | \( 1 + (1.71 - 1.03i)T + (0.468 - 0.883i)T^{2} \) |
| 73 | \( 1 + (0.725 - 0.687i)T^{2} \) |
| 79 | \( 1 + (-0.947 + 0.319i)T + (0.796 - 0.605i)T^{2} \) |
| 83 | \( 1 + (0.370 + 0.928i)T^{2} \) |
| 89 | \( 1 + (-0.647 - 0.762i)T^{2} \) |
| 97 | \( 1 + (0.725 + 0.687i)T^{2} \) |
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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
−8.904002686746150143039648940193, −8.267103434965461917494658332061, −7.30035994352926043356001965809, −6.36723743422722790026311991955, −5.88906537051171580616328696002, −5.32445119537300393111100181410, −3.98298000374475051641724776416, −3.01518361368731930635411564877, −2.34220998619901100547215421277, −1.69082957410604514727766461662,
1.35272216614411083307350701628, 2.26437484839307484112448612627, 3.10432100615457618352328521119, 3.74767357878982172209863838179, 4.93690550792149623226647539735, 5.69393984384332756783128635599, 6.80381259871860466973075404473, 7.44574261141511593352004592531, 7.74666810177202173205258924339, 8.738458391747015765754317593733