L(s) = 1 | + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + 1.17i·7-s + (0.809 + 0.587i)9-s + (−1.30 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (1.80 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + (−0.690 − 0.951i)23-s + (0.309 + 0.951i)25-s + (−1.11 − 0.363i)28-s + (0.690 − 0.951i)35-s + (−0.809 + 0.587i)36-s − 1.90i·43-s + (−0.499 − 1.53i)44-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + 1.17i·7-s + (0.809 + 0.587i)9-s + (−1.30 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (1.80 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + (−0.690 − 0.951i)23-s + (0.309 + 0.951i)25-s + (−1.11 − 0.363i)28-s + (0.690 − 0.951i)35-s + (−0.809 + 0.587i)36-s − 1.90i·43-s + (−0.499 − 1.53i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6948572259\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6948572259\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - 1.17iT - T^{2} \) |
| 11 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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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
−11.87933652198433878513471069887, −10.40907209332789332122411361977, −9.608214347272229386734718227830, −8.477187151466016427696281918093, −7.83727882344111302115200046105, −7.31725336482532786333466471352, −5.44531631168305818233522189093, −4.77961422110777909717812672801, −3.60526783870640191779209270231, −2.30564430437665130897188338848,
0.978776212574264106031507618886, 3.22470335007746915705417134622, 4.15242105935830573817782920367, 5.34524417964202535976329559971, 6.41691272283075467280982156779, 7.47340117282881315843328996968, 8.059135648538837434917684708481, 9.557536894224856921804389280368, 10.28266028916752015184166060724, 10.78612107398408201846071956577