L(s) = 1 | − 1.54·2-s + 3-s + 0.384·4-s + 2.83·5-s − 1.54·6-s + 2.51·7-s + 2.49·8-s + 9-s − 4.37·10-s − 11-s + 0.384·12-s − 3.88·14-s + 2.83·15-s − 4.62·16-s − 3.71·17-s − 1.54·18-s + 1.44·19-s + 1.09·20-s + 2.51·21-s + 1.54·22-s + 2.70·23-s + 2.49·24-s + 3.03·25-s + 27-s + 0.968·28-s + 3.96·29-s − 4.37·30-s + ⋯ |
L(s) = 1 | − 1.09·2-s + 0.577·3-s + 0.192·4-s + 1.26·5-s − 0.630·6-s + 0.950·7-s + 0.881·8-s + 0.333·9-s − 1.38·10-s − 0.301·11-s + 0.111·12-s − 1.03·14-s + 0.731·15-s − 1.15·16-s − 0.901·17-s − 0.363·18-s + 0.330·19-s + 0.244·20-s + 0.548·21-s + 0.329·22-s + 0.564·23-s + 0.509·24-s + 0.607·25-s + 0.192·27-s + 0.182·28-s + 0.735·29-s − 0.799·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.903846403\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.903846403\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 - 0.0964T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 9.76T + 41T^{2} \) |
| 43 | \( 1 + 0.467T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 0.316T + 53T^{2} \) |
| 59 | \( 1 + 7.82T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 0.361T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 0.450T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.444130226089709999628019706094, −7.60388400145377643878939337661, −6.99203371587675688651066232638, −6.15552815204387053488402695635, −5.04344806165255955793200410999, −4.77571979866156480147227948443, −3.54163819942530981346174879840, −2.27566874894163142083154046243, −1.88229516120033683396580237374, −0.884191853105487492503187174502,
0.884191853105487492503187174502, 1.88229516120033683396580237374, 2.27566874894163142083154046243, 3.54163819942530981346174879840, 4.77571979866156480147227948443, 5.04344806165255955793200410999, 6.15552815204387053488402695635, 6.99203371587675688651066232638, 7.60388400145377643878939337661, 8.444130226089709999628019706094