| L(s) = 1 | + (−1 + 2i)5-s + i·7-s + 3·9-s − 4i·13-s − 4i·17-s − 4·19-s − 8i·23-s + (−3 − 4i)25-s + 2·29-s + 8·31-s + (−2 − i)35-s − 8i·37-s + 6·41-s + 8i·43-s + (−3 + 6i)45-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s + 0.377i·7-s + 9-s − 1.10i·13-s − 0.970i·17-s − 0.917·19-s − 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s + 1.43·31-s + (−0.338 − 0.169i)35-s − 1.31i·37-s + 0.937·41-s + 1.21i·43-s + (−0.447 + 0.894i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596512071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596512071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
| 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
−8.947464972551267269049887564185, −8.037338517965774741813117061386, −7.51161343000160009355030276595, −6.60028161874464442817477133929, −6.08328048998139571672241179349, −4.79862045688158113138469808783, −4.18776467257530117318188829476, −2.99111829478864071898903528864, −2.37069313663838171317632311511, −0.65356668859271714552366025625,
1.11125498281815123418165821986, 1.97587355382782275594029392529, 3.65576844231558436063603662451, 4.23292128106918280694979826804, 4.88100801294747354385098416517, 6.00126502982397296958586899017, 6.86748832236194860180600193931, 7.54498922559323397897075835291, 8.410911655944872278933149410750, 8.955619885332357417309218444663
