| L(s) = 1 | − 2.23i·5-s + 0.167i·7-s + 11.8i·11-s + 9.42·13-s − 0.812i·17-s + 12.4i·19-s + (−11.0 + 20.1i)23-s − 5.00·25-s − 30.8·29-s + 31.2·31-s + 0.375·35-s + 0.0361i·37-s + 11.4·41-s − 59.8i·43-s − 35.5·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.0239i·7-s + 1.07i·11-s + 0.725·13-s − 0.0477i·17-s + 0.652i·19-s + (−0.478 + 0.878i)23-s − 0.200·25-s − 1.06·29-s + 1.00·31-s + 0.0107·35-s + 0.000976i·37-s + 0.279·41-s − 1.39i·43-s − 0.756·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6981154443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6981154443\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (11.0 - 20.1i)T \) |
| good | 7 | \( 1 - 0.167iT - 49T^{2} \) |
| 11 | \( 1 - 11.8iT - 121T^{2} \) |
| 13 | \( 1 - 9.42T + 169T^{2} \) |
| 17 | \( 1 + 0.812iT - 289T^{2} \) |
| 19 | \( 1 - 12.4iT - 361T^{2} \) |
| 29 | \( 1 + 30.8T + 841T^{2} \) |
| 31 | \( 1 - 31.2T + 961T^{2} \) |
| 37 | \( 1 - 0.0361iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 59.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 94.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 82.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 32.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 3.42iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 105.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 118.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 42.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 120. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 6.47iT - 9.40e3T^{2} \) |
| 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
−8.723985674582301365308708562463, −7.55431167619338650928605811616, −7.44213122285288674435446418541, −6.15455174342741163043928872524, −5.73573300673025855823682493288, −4.70574290737239292634735678996, −4.10359591230838435608104380673, −3.20914916713779257624172974896, −2.01011160133503711682955535298, −1.28709488755573357373510272161,
0.14579592521021215054637477937, 1.26120828467569726346604646314, 2.48428253082640316491507124124, 3.24678668819551357753179255777, 4.04009832007149048787173044149, 4.93725757341267415548685608669, 5.95882922933887100091246557847, 6.32815407813895257682132617051, 7.17593285090065930776727597180, 8.075072398329681872667560258623
