L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.965 − 0.258i)5-s + i·7-s + (0.366 + 1.36i)10-s + 1.41i·11-s + (−0.866 − 0.5i)13-s + (1.22 − 0.707i)14-s + (0.499 + 0.866i)16-s + (0.707 + 1.22i)17-s − 19-s + (0.707 − 0.707i)20-s + (1.73 − 1.00i)22-s + (0.866 + 0.499i)25-s + 1.41i·26-s + ⋯ |
L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.965 − 0.258i)5-s + i·7-s + (0.366 + 1.36i)10-s + 1.41i·11-s + (−0.866 − 0.5i)13-s + (1.22 − 0.707i)14-s + (0.499 + 0.866i)16-s + (0.707 + 1.22i)17-s − 19-s + (0.707 − 0.707i)20-s + (1.73 − 1.00i)22-s + (0.866 + 0.499i)25-s + 1.41i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3667658323\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3667658323\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−10.43030405854688495854478802016, −9.727055695705269726969095478708, −8.918951337781521191380462123710, −8.187897451676408901740628055259, −7.40218331243360735577796937551, −6.07046243299068854560316857759, −4.87878664907020832849186416168, −3.84812554303022861797737748169, −2.70598970899494006003141447046, −1.71119170890102580854958029434,
0.46123738679950155791205104136, 3.00460355998707593298031223932, 4.05592640012137280057023630941, 5.22842945130862481020619507881, 6.32345515182801617317624351594, 7.20690757745263006479330070119, 7.56498920893636604994031239783, 8.438863300036306950402614837487, 9.183419976794329123620264262148, 10.15495799390014303916878404023