| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 2·11-s + 12-s − 14-s + 16-s + 5·17-s − 18-s + 21-s − 2·22-s + 23-s − 24-s − 5·25-s + 27-s + 28-s − 29-s + 7·31-s − 32-s + 2·33-s − 5·34-s + 36-s − 37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.218·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 25-s + 0.192·27-s + 0.188·28-s − 0.185·29-s + 1.25·31-s − 0.176·32-s + 0.348·33-s − 0.857·34-s + 1/6·36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.69537071251826, −12.16974344593048, −11.78983088111222, −11.53890689769667, −10.88566708015836, −10.35760990753042, −9.909433511875368, −9.694676581060100, −9.132180071768804, −8.566115170628389, −8.284972255010091, −7.770614142790854, −7.494872842014999, −6.691485869549216, −6.572517116864640, −5.802543975696238, −5.336745085767715, −4.757875561586773, −4.156221195349889, −3.550840875288014, −3.180329443534131, −2.565653061260860, −1.842631082800887, −1.470098986257668, −0.8802943850605622, 0,
0.8802943850605622, 1.470098986257668, 1.842631082800887, 2.565653061260860, 3.180329443534131, 3.550840875288014, 4.156221195349889, 4.757875561586773, 5.336745085767715, 5.802543975696238, 6.572517116864640, 6.691485869549216, 7.494872842014999, 7.770614142790854, 8.284972255010091, 8.566115170628389, 9.132180071768804, 9.694676581060100, 9.909433511875368, 10.35760990753042, 10.88566708015836, 11.53890689769667, 11.78983088111222, 12.16974344593048, 12.69537071251826
