L(s) = 1 | + (−0.222 + 0.974i)4-s + (0.0931 − 1.24i)5-s + (0.988 + 1.71i)7-s + (−0.733 + 0.680i)9-s + (−0.0332 − 0.145i)11-s + (−0.900 − 0.433i)16-s + (0.0546 + 0.728i)17-s + (−0.733 − 0.680i)19-s + (1.19 + 0.367i)20-s + (1.57 + 0.487i)23-s + (−0.548 − 0.0827i)25-s + (−1.88 + 0.582i)28-s + (2.22 − 1.07i)35-s + (−0.5 − 0.866i)36-s + 43-s + 0.149·44-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)4-s + (0.0931 − 1.24i)5-s + (0.988 + 1.71i)7-s + (−0.733 + 0.680i)9-s + (−0.0332 − 0.145i)11-s + (−0.900 − 0.433i)16-s + (0.0546 + 0.728i)17-s + (−0.733 − 0.680i)19-s + (1.19 + 0.367i)20-s + (1.57 + 0.487i)23-s + (−0.548 − 0.0827i)25-s + (−1.88 + 0.582i)28-s + (2.22 − 1.07i)35-s + (−0.5 − 0.866i)36-s + 43-s + 0.149·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9691352381\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9691352381\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + (0.733 + 0.680i)T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 3 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 5 | \( 1 + (-0.0931 + 1.24i)T + (-0.988 - 0.149i)T^{2} \) |
| 7 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 17 | \( 1 + (-0.0546 - 0.728i)T + (-0.988 + 0.149i)T^{2} \) |
| 23 | \( 1 + (-1.57 - 0.487i)T + (0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 31 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 71 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 73 | \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 + (0.900 - 0.433i)T^{2} \) |
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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
−10.84006026122347162800805212842, −9.142196175121118066452420188677, −8.742019875106287840895018392646, −8.414897596130060180734866176695, −7.47123690440304396531014204951, −5.90601774059150855234454049801, −5.11888382235018037564269467429, −4.55664581382038860756969482343, −2.96042946561219285416612142635, −1.94836381140126406687420499517,
1.11995656962691323834649655472, 2.73015948272330959776428375001, 4.00127865347870762904306307810, 4.89487247650206022151892315798, 6.07503491550190610470745963772, 6.86906918134381757196668333040, 7.50712747116519508695659227227, 8.691154689556512616272571825774, 9.693699503681568388460205466005, 10.54010152728688692217142859904