| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s − 4·14-s + 15-s + 16-s + 2·17-s − 18-s − 20-s − 4·21-s + 2·22-s + 23-s + 24-s + 25-s − 27-s + 4·28-s − 4·29-s − 30-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.872·21-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.755·28-s − 0.742·29-s − 0.182·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9529657857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9529657857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−10.73818758737342003208652801274, −9.643707358459316204749629879212, −8.633322632355962728976135385188, −7.76597288344800187250732468158, −7.35825771010007829981691032373, −5.95675854509685083992398916112, −5.10595242146171806051909461506, −4.07854085227722571071204587801, −2.39291634987828873969007813256, −0.987830286047861132644615391405,
0.987830286047861132644615391405, 2.39291634987828873969007813256, 4.07854085227722571071204587801, 5.10595242146171806051909461506, 5.95675854509685083992398916112, 7.35825771010007829981691032373, 7.76597288344800187250732468158, 8.633322632355962728976135385188, 9.643707358459316204749629879212, 10.73818758737342003208652801274
