L(s) = 1 | + 2.30·2-s − 1.84i·3-s + 3.30·4-s − 1.41i·5-s − 4.24i·6-s + 7-s + 3.00·8-s − 0.394·9-s − 3.25i·10-s + (−3 − 1.41i)11-s − 6.08i·12-s + 6.51i·13-s + 2.30·14-s − 2.60·15-s + 0.302·16-s + 0.428i·17-s + ⋯ |
L(s) = 1 | + 1.62·2-s − 1.06i·3-s + 1.65·4-s − 0.632i·5-s − 1.73i·6-s + 0.377·7-s + 1.06·8-s − 0.131·9-s − 1.02i·10-s + (−0.904 − 0.426i)11-s − 1.75i·12-s + 1.80i·13-s + 0.615·14-s − 0.672·15-s + 0.0756·16-s + 0.103i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.83396 - 1.82479i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.83396 - 1.82479i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (3 + 1.41i)T \) |
| 43 | \( 1 + (-5 - 4.24i)T \) |
good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.84iT - 3T^{2} \) |
| 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 6.51iT - 13T^{2} \) |
| 17 | \( 1 - 0.428iT - 17T^{2} \) |
| 19 | \( 1 + 0.394T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 7.60T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 7.49iT - 37T^{2} \) |
| 41 | \( 1 + 3.81iT - 41T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 - 9.34iT - 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.28iT - 79T^{2} \) |
| 83 | \( 1 + 8.05iT - 83T^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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
−11.58825314993409844546174214037, −10.22560563823105505426948089585, −8.779629868626009803669525852982, −7.88268717596108236794608249366, −6.74871269037001321900843866825, −6.20940774178026233671834650928, −4.91389680308415909090274990372, −4.34284097890172052390037640476, −2.76490013759328441427785454614, −1.59016805892675114368644140790,
2.63708582034311950503944509241, 3.39360493569368124012090627695, 4.57796859312646286972980107839, 5.14262427933254171367005034606, 6.10430878515351726136082869449, 7.28410004351603991045961480492, 8.298209731794765899443658446063, 9.878225745655592704943948548336, 10.52525388580504790874328219455, 11.08610575302526646689280636620