L(s) = 1 | + (0.540 − 0.841i)2-s + (−0.755 + 0.654i)3-s + (−0.415 − 0.909i)4-s + (0.959 + 0.281i)5-s + (0.142 + 0.989i)6-s + (0.989 + 0.857i)7-s + (−0.989 − 0.142i)8-s + (0.142 − 0.989i)9-s + (0.755 − 0.654i)10-s + (0.909 + 0.415i)12-s + (1.25 − 0.368i)14-s + (−0.909 + 0.415i)15-s + (−0.654 + 0.755i)16-s + (−0.755 − 0.654i)18-s + (−0.142 − 0.989i)20-s − 1.30·21-s + ⋯ |
L(s) = 1 | + (0.540 − 0.841i)2-s + (−0.755 + 0.654i)3-s + (−0.415 − 0.909i)4-s + (0.959 + 0.281i)5-s + (0.142 + 0.989i)6-s + (0.989 + 0.857i)7-s + (−0.989 − 0.142i)8-s + (0.142 − 0.989i)9-s + (0.755 − 0.654i)10-s + (0.909 + 0.415i)12-s + (1.25 − 0.368i)14-s + (−0.909 + 0.415i)15-s + (−0.654 + 0.755i)16-s + (−0.755 − 0.654i)18-s + (−0.142 − 0.989i)20-s − 1.30·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.355808652\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.355808652\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.540 + 0.841i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (0.540 - 0.841i)T \) |
good | 7 | \( 1 + (-0.989 - 0.857i)T + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.512 - 0.234i)T + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.425 + 1.45i)T + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (-0.822 + 0.118i)T + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + 1.91iT - T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (0.153 - 0.239i)T + (-0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (1.89 - 0.557i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 + 0.540i)T^{2} \) |
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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
−9.901510587358744141552536830328, −9.250314303145792431324526581125, −8.525860216285868547225720271886, −6.97879698495905737197829550239, −5.89130614980937521180754022403, −5.50935923210155711916895428022, −4.78728896597494815706438001293, −3.74480944697032341135449170993, −2.53867386021650809110801921341, −1.53217134780364078173070908754,
1.29752381764834689714350010580, 2.65122092348204690053227880031, 4.47554509475346937654177969641, 4.74114313777862577762221068563, 5.92189957945883621384034416223, 6.29659626532875994926913911596, 7.34221347372059315765764716166, 7.901514298567160905356863894017, 8.725386290269144565307782249387, 9.829319572403538954089719105340