L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 3·11-s + 12-s + 13-s − 14-s + 3·15-s + 16-s − 3·17-s − 18-s − 7·19-s + 3·20-s + 21-s − 3·22-s + 9·23-s − 24-s + 4·25-s − 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.60·19-s + 0.670·20-s + 0.218·21-s − 0.639·22-s + 1.87·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690956649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690956649\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.78600737924430448083487191522, −9.626606150401529001958201314809, −9.079799022404556504658251347602, −8.505529332786924267956180509594, −7.16011708962485875279239287487, −6.45501714883769400197534752234, −5.38516927860402892091834425845, −3.95823828965663710949182273306, −2.41262911451317421726904251873, −1.54544973372056719849266422970,
1.54544973372056719849266422970, 2.41262911451317421726904251873, 3.95823828965663710949182273306, 5.38516927860402892091834425845, 6.45501714883769400197534752234, 7.16011708962485875279239287487, 8.505529332786924267956180509594, 9.079799022404556504658251347602, 9.626606150401529001958201314809, 10.78600737924430448083487191522