| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 2·7-s + 8-s + 6·9-s + 11-s − 3·12-s − 2·14-s + 16-s + 3·17-s + 6·18-s + 6·21-s + 22-s + 5·23-s − 3·24-s − 5·25-s − 9·27-s − 2·28-s + 7·29-s + 4·31-s + 32-s − 3·33-s + 3·34-s + 6·36-s − 37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s + 0.301·11-s − 0.866·12-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.41·18-s + 1.30·21-s + 0.213·22-s + 1.04·23-s − 0.612·24-s − 25-s − 1.73·27-s − 0.377·28-s + 1.29·29-s + 0.718·31-s + 0.176·32-s − 0.522·33-s + 0.514·34-s + 36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 223 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.31015772735579, −14.85272499649880, −13.94881531568392, −13.47347727371622, −13.10512865385407, −12.28618901127252, −12.12679757510141, −11.70213620676388, −11.12083879930516, −10.51319999013916, −10.03225125485721, −9.723429622936790, −8.774510242358393, −8.033215656168371, −7.262536521320555, −6.663536271213174, −6.454377946518338, −5.823359642726368, −5.196604833527621, −4.855832563937315, −4.098287712328052, −3.443743947560786, −2.761606847556948, −1.623829197962608, −0.9488822453946294, 0,
0.9488822453946294, 1.623829197962608, 2.761606847556948, 3.443743947560786, 4.098287712328052, 4.855832563937315, 5.196604833527621, 5.823359642726368, 6.454377946518338, 6.663536271213174, 7.262536521320555, 8.033215656168371, 8.774510242358393, 9.723429622936790, 10.03225125485721, 10.51319999013916, 11.12083879930516, 11.70213620676388, 12.12679757510141, 12.28618901127252, 13.10512865385407, 13.47347727371622, 13.94881531568392, 14.85272499649880, 15.31015772735579
