| L(s) = 1 | − i·2-s − 4-s + 4·5-s + 7-s + i·8-s − 4i·10-s − 5i·11-s − 13-s − i·14-s + 16-s + 2i·17-s − i·19-s − 4·20-s − 5·22-s + 23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.78·5-s + 0.377·7-s + 0.353i·8-s − 1.26i·10-s − 1.50i·11-s − 0.277·13-s − 0.267i·14-s + 0.250·16-s + 0.485i·17-s − 0.229i·19-s − 0.894·20-s − 1.06·22-s + 0.208·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.500299870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.500299870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 + (5 + 2i)T \) |
| good | 5 | \( 1 - 4T + 5T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
| 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
−8.509856182947947856165146535538, −7.76607466250321713996276783752, −6.59254369422427419279060298915, −5.74363864478352164779678251273, −5.56983300410448900868099788985, −4.45309676060584793082531222732, −3.43111730382037521268894486502, −2.47389121392009059671106644378, −1.84281819775291011579260069246, −0.72533349942862686090242301488,
1.42845250353799179949477817195, 2.10191307280251548419552990276, 3.21184076511002297502060769383, 4.68897684148688601161253376848, 5.01780853996702247676875529556, 5.75009127488683462969562615599, 6.69390225430835546287573339799, 7.01617055520380168210634948251, 7.961403743963443736824398926331, 8.879759510491198384649807138671
