L(s) = 1 | + (0.0980 + 0.995i)2-s + (−0.980 + 0.195i)4-s + (0.195 − 0.980i)5-s + (−0.222 + 0.536i)7-s + (−0.290 − 0.956i)8-s + (−0.382 − 0.923i)9-s + (0.995 + 0.0980i)10-s + (−0.831 + 0.555i)11-s + (0.247 + 1.24i)13-s + (−0.555 − 0.168i)14-s + (0.923 − 0.382i)16-s + (−0.666 + 0.666i)17-s + (0.881 − 0.471i)18-s + i·20-s + (−0.634 − 0.773i)22-s + ⋯ |
L(s) = 1 | + (0.0980 + 0.995i)2-s + (−0.980 + 0.195i)4-s + (0.195 − 0.980i)5-s + (−0.222 + 0.536i)7-s + (−0.290 − 0.956i)8-s + (−0.382 − 0.923i)9-s + (0.995 + 0.0980i)10-s + (−0.831 + 0.555i)11-s + (0.247 + 1.24i)13-s + (−0.555 − 0.168i)14-s + (0.923 − 0.382i)16-s + (−0.666 + 0.666i)17-s + (0.881 − 0.471i)18-s + i·20-s + (−0.634 − 0.773i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7263149118\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7263149118\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0980 - 0.995i)T \) |
| 5 | \( 1 + (-0.195 + 0.980i)T \) |
| 11 | \( 1 + (0.831 - 0.555i)T \) |
good | 3 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.536i)T + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.247 - 1.24i)T + (-0.923 + 0.382i)T^{2} \) |
| 17 | \( 1 + (0.666 - 0.666i)T - iT^{2} \) |
| 19 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 - 1.66iT - T^{2} \) |
| 37 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-1.10 - 1.65i)T + (-0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 59 | \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.0750 - 0.181i)T + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + (1.24 - 0.247i)T + (0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (1.81 + 0.750i)T + (0.707 + 0.707i)T^{2} \) |
| 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
−8.813815142037929944027030944964, −8.558889065587913242657413944978, −7.55964312722259626100066494166, −6.69709229367137536170113803992, −6.10334915664328358498261962413, −5.42590851047640746798341928321, −4.57314978184906069371384022793, −4.01642461370335490989057931794, −2.78014653332554751652298743422, −1.40018568343111515626966917206,
0.41617375716728551864830525526, 2.18133419076696237632011671849, 2.74517173323142675600519245611, 3.48640240862141362826807337976, 4.43061705315332437330846254350, 5.51893932724620066908619387726, 5.81906849337050645252095862069, 7.12170215968501143685562587492, 7.81928214315652968073840550211, 8.422386399731663894871237102354