| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 3·11-s + 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 2·19-s − 21-s − 3·22-s − 3·23-s − 24-s − 5·25-s + 26-s + 27-s − 28-s + 8·31-s − 32-s + 3·33-s + 34-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 + 0.904·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.218·21-s − 0.639·22-s − 0.625·23-s − 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.43·31-s − 0.176·32-s + 0.522·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.06049487108884, −13.80132504096237, −13.20574229015777, −12.73491583727723, −11.85535415261396, −11.77587350618001, −11.38989095733239, −10.27405350979500, −10.11277116779290, −9.762835767441115, −9.147791501033433, −8.579107231493292, −8.254598162762188, −7.719650044257398, −7.026686207882162, −6.553901911522726, −6.306902637832533, −5.315972506541017, −4.888294836028304, −3.930604927375308, −3.596963086174264, −2.973684296813966, −2.147400188144107, −1.734356970877373, −0.8952826300806512, 0,
0.8952826300806512, 1.734356970877373, 2.147400188144107, 2.973684296813966, 3.596963086174264, 3.930604927375308, 4.888294836028304, 5.315972506541017, 6.306902637832533, 6.553901911522726, 7.026686207882162, 7.719650044257398, 8.254598162762188, 8.579107231493292, 9.147791501033433, 9.762835767441115, 10.11277116779290, 10.27405350979500, 11.38989095733239, 11.77587350618001, 11.85535415261396, 12.73491583727723, 13.20574229015777, 13.80132504096237, 14.06049487108884
