L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)11-s + (0.809 + 0.587i)12-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s − 0.999·18-s + (−1.80 + 0.587i)19-s + (0.809 + 0.587i)22-s + 24-s + (−0.309 − 0.951i)25-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)11-s + (0.809 + 0.587i)12-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s − 0.999·18-s + (−1.80 + 0.587i)19-s + (0.809 + 0.587i)22-s + 24-s + (−0.309 − 0.951i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9878336230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9878336230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
good | 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 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 + 1.17iT - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 1.17iT - T^{2} \) |
| 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
−12.11760008892500796427656830928, −11.23219503082205516994490223784, −10.36469697255195961855380329691, −9.715693376172804525377898265638, −8.598551555791422644854415597380, −6.79997513346974898580798213337, −5.86363959105077971020893479102, −4.60400992516194197325476296894, −3.99531835212255375205442183375, −2.37362098857690372097412503603,
2.34333803514186724896722920634, 3.90665421499112695506344499848, 5.34806349899617237912977398325, 6.28867917791229953829435959466, 6.99820737920006169748928740343, 8.183784225832969237458163601321, 8.863944082603331619986402463646, 10.88320607462122253728744546217, 11.42029206930293426009731281449, 12.53754645894466774899537708340