| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s − 12-s + 2·13-s − 15-s − 16-s + 5·17-s + 18-s + 5·19-s + 20-s − 3·23-s − 3·24-s + 25-s + 2·26-s + 27-s + 29-s − 30-s − 8·31-s + 5·32-s + 5·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.625·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.185·29-s − 0.182·30-s − 1.43·31-s + 0.883·32-s + 0.857·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.637858090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.637858090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−13.95472170634152, −13.49739522754905, −12.87681949791871, −12.49268013599854, −12.16407921627741, −11.42222145852647, −11.16031859210017, −10.23604244411336, −9.888411574922450, −9.327499272912668, −8.894979917647281, −8.320933662169308, −7.833492873530708, −7.402564630649888, −6.741317112086571, −6.019552609984809, −5.453239845702796, −5.170807461216981, −4.342644237128603, −3.753803912209453, −3.538425197610821, −2.924799553855265, −2.169367114341498, −1.223926979241117, −0.5783511040264241,
0.5783511040264241, 1.223926979241117, 2.169367114341498, 2.924799553855265, 3.538425197610821, 3.753803912209453, 4.342644237128603, 5.170807461216981, 5.453239845702796, 6.019552609984809, 6.741317112086571, 7.402564630649888, 7.833492873530708, 8.320933662169308, 8.894979917647281, 9.327499272912668, 9.888411574922450, 10.23604244411336, 11.16031859210017, 11.42222145852647, 12.16407921627741, 12.49268013599854, 12.87681949791871, 13.49739522754905, 13.95472170634152
