L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 2·14-s + 15-s + 16-s − 18-s − 2·19-s − 20-s − 2·21-s − 6·23-s + 24-s + 25-s − 26-s − 27-s + 2·28-s − 30-s + 4·31-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 + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9077584843\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9077584843\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.59054683597388, −11.88368221899400, −11.69195512694414, −11.15961900938241, −10.90640791730912, −10.34788274182098, −9.893622387317465, −9.545493174604731, −8.775706691919451, −8.457672725461646, −8.038847979451531, −7.612167415041125, −7.100031077705222, −6.625177687598421, −6.016454386870372, −5.748861935974418, −5.049323904898708, −4.490392361754123, −4.111024865529592, −3.523811990968884, −2.768643821286784, −2.188171389589747, −1.597361004062226, −1.032771756039637, −0.3294551865737045,
0.3294551865737045, 1.032771756039637, 1.597361004062226, 2.188171389589747, 2.768643821286784, 3.523811990968884, 4.111024865529592, 4.490392361754123, 5.049323904898708, 5.748861935974418, 6.016454386870372, 6.625177687598421, 7.100031077705222, 7.612167415041125, 8.038847979451531, 8.457672725461646, 8.775706691919451, 9.545493174604731, 9.893622387317465, 10.34788274182098, 10.90640791730912, 11.15961900938241, 11.69195512694414, 11.88368221899400, 12.59054683597388