L(s) = 1 | + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (−1.95 − 0.389i)5-s + (0.382 + 0.923i)8-s + (0.130 + 0.991i)9-s + (1.78 − 0.882i)10-s + (0.793 − 0.608i)13-s + (−0.866 − 0.499i)16-s + (0.965 − 0.258i)17-s + (−0.707 − 0.707i)18-s + (−0.882 + 1.78i)20-s + (2.75 + 1.14i)25-s + (−0.258 + 0.965i)26-s + (0.882 − 0.0578i)29-s + (0.991 − 0.130i)32-s + ⋯ |
L(s) = 1 | + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (−1.95 − 0.389i)5-s + (0.382 + 0.923i)8-s + (0.130 + 0.991i)9-s + (1.78 − 0.882i)10-s + (0.793 − 0.608i)13-s + (−0.866 − 0.499i)16-s + (0.965 − 0.258i)17-s + (−0.707 − 0.707i)18-s + (−0.882 + 1.78i)20-s + (2.75 + 1.14i)25-s + (−0.258 + 0.965i)26-s + (0.882 − 0.0578i)29-s + (0.991 − 0.130i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5105753335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5105753335\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.793 - 0.608i)T \) |
| 13 | \( 1 + (-0.793 + 0.608i)T \) |
| 17 | \( 1 + (-0.965 + 0.258i)T \) |
good | 3 | \( 1 + (-0.130 - 0.991i)T^{2} \) |
| 5 | \( 1 + (1.95 + 0.389i)T + (0.923 + 0.382i)T^{2} \) |
| 7 | \( 1 + (0.130 - 0.991i)T^{2} \) |
| 11 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 19 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 23 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 29 | \( 1 + (-0.882 + 0.0578i)T + (0.991 - 0.130i)T^{2} \) |
| 31 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-1.50 + 0.0983i)T + (0.991 - 0.130i)T^{2} \) |
| 41 | \( 1 + (-0.608 - 0.206i)T + (0.793 + 0.608i)T^{2} \) |
| 43 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.241 - 0.0999i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 61 | \( 1 + (1.31 + 0.0862i)T + (0.991 + 0.130i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 73 | \( 1 + (-0.369 + 1.85i)T + (-0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.05 - 0.357i)T + (0.793 - 0.608i)T^{2} \) |
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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
−10.55470533260027947526825396671, −9.349049939650049044428293108490, −8.422005343437679564591130970861, −7.78007363078432282396051638162, −7.56584713013871240675406265375, −6.26640970115232261642099100109, −5.10792687689962175911652994640, −4.32996759377928015455564543040, −3.01728134921855214849962639074, −1.01720413761775141385137351185,
0.969399870188528542905061957698, 2.98185995064489374579029723067, 3.73371503552015890613273920740, 4.37154343603054024531774403726, 6.34610109180698555172639850286, 7.11341366780422025495092752682, 7.927298007446046561339862633672, 8.514534757419022262830166865894, 9.383986480901261918934048753230, 10.39185820752908827285247630158