| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 5·11-s + 12-s + 14-s + 15-s + 16-s − 3·17-s − 18-s + 19-s + 20-s − 21-s + 5·22-s + 3·23-s − 24-s − 4·25-s + 27-s − 28-s + 9·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s + 1.06·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481919160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481919160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.88423413841871506203636604586, −7.53102487461663975706817859964, −6.61270975441273816602997295472, −6.01853701822204108385590735367, −5.14408356509184259794282746045, −4.37078468584442969243737745111, −3.19242539500705319758808535486, −2.63163126271274006390959216123, −1.91688586890288739737385366338, −0.64494925744861723051140272143,
0.64494925744861723051140272143, 1.91688586890288739737385366338, 2.63163126271274006390959216123, 3.19242539500705319758808535486, 4.37078468584442969243737745111, 5.14408356509184259794282746045, 6.01853701822204108385590735367, 6.61270975441273816602997295472, 7.53102487461663975706817859964, 7.88423413841871506203636604586
