| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s + 9-s − 3·10-s − 11-s − 12-s + 13-s + 3·15-s + 16-s − 3·17-s + 18-s + 19-s − 3·20-s − 22-s − 23-s − 24-s + 4·25-s + 26-s − 27-s + 5·29-s + 3·30-s − 6·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s − 0.670·20-s − 0.213·22-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.928·29-s + 0.547·30-s − 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529846118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529846118\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.254386201443168436263551057727, −7.65241658576590353597599492216, −6.94539066562319255877064295322, −6.27888711821180087770042404568, −5.35954031898323501889962460397, −4.65939875152174409378503106898, −3.96129681331778747711242209548, −3.28722635591842678016893069786, −2.11142405647682177968306732752, −0.64801473181796500221977585357,
0.64801473181796500221977585357, 2.11142405647682177968306732752, 3.28722635591842678016893069786, 3.96129681331778747711242209548, 4.65939875152174409378503106898, 5.35954031898323501889962460397, 6.27888711821180087770042404568, 6.94539066562319255877064295322, 7.65241658576590353597599492216, 8.254386201443168436263551057727
