L(s) = 1 | + (0.149 − 0.360i)2-s + (0.599 + 0.599i)4-s + (−0.555 + 0.831i)5-s + (0.666 − 0.275i)8-s + (0.216 + 0.324i)10-s + 0.566i·16-s + (−0.980 + 0.195i)17-s + (0.707 + 0.292i)19-s + (−0.831 + 0.165i)20-s + (1.38 + 0.275i)23-s + (−0.382 − 0.923i)25-s + (−1.08 + 0.216i)31-s + (0.870 + 0.360i)32-s + (−0.0761 + 0.382i)34-s + (0.211 − 0.211i)38-s + ⋯ |
L(s) = 1 | + (0.149 − 0.360i)2-s + (0.599 + 0.599i)4-s + (−0.555 + 0.831i)5-s + (0.666 − 0.275i)8-s + (0.216 + 0.324i)10-s + 0.566i·16-s + (−0.980 + 0.195i)17-s + (0.707 + 0.292i)19-s + (−0.831 + 0.165i)20-s + (1.38 + 0.275i)23-s + (−0.382 − 0.923i)25-s + (−1.08 + 0.216i)31-s + (0.870 + 0.360i)32-s + (−0.0761 + 0.382i)34-s + (0.211 − 0.211i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.110294647\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110294647\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.555 - 0.831i)T \) |
| 17 | \( 1 + (0.980 - 0.195i)T \) |
good | 2 | \( 1 + (-0.149 + 0.360i)T + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-1.38 - 0.275i)T + (0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (1.08 - 0.216i)T + (0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (1.17 + 1.17i)T + iT^{2} \) |
| 53 | \( 1 + (-0.750 + 1.81i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-1.81 - 0.750i)T + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.382 + 0.923i)T^{2} \) |
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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
−10.88137661778045858468578817577, −10.00596416406809220107507156621, −8.817059504987637335439445258714, −7.902436304625418317624489987837, −7.07542333782128408651290389316, −6.56849147095198230860365947694, −5.11671105743988763211051221270, −3.83110626620090152401978094844, −3.17451362025074767724300834827, −2.01592566589998546916894904115,
1.29638118279247382254072612825, 2.82012841196487127354143360108, 4.35044469541934666909363274080, 5.08978721103483715784881114472, 6.00256515611371438780367055279, 7.09373343240269684235745892691, 7.64934920828001870299166952659, 8.856305725151448354048793704220, 9.412119494162438673174950567117, 10.66977553297740007510348670175