| L(s) = 1 | + 3-s − 0.561i·5-s + i·7-s + 9-s − 1.43i·11-s + (−0.561 + 3.56i)13-s − 0.561i·15-s + 5.68·17-s − 2.56i·19-s + i·21-s − 5.68·23-s + 4.68·25-s + 27-s − 2.56·29-s − 10.2i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.251i·5-s + 0.377i·7-s + 0.333·9-s − 0.433i·11-s + (−0.155 + 0.987i)13-s − 0.144i·15-s + 1.37·17-s − 0.587i·19-s + 0.218i·21-s − 1.18·23-s + 0.936·25-s + 0.192·27-s − 0.475·29-s − 1.84i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.502068152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502068152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (0.561 - 3.56i)T \) |
| good | 5 | \( 1 + 0.561iT - 5T^{2} \) |
| 11 | \( 1 + 1.43iT - 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + 2.56iT - 19T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 + 1.68iT - 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.24iT - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 2.56T + 61T^{2} \) |
| 67 | \( 1 + 7.12iT - 67T^{2} \) |
| 71 | \( 1 - 15.3iT - 71T^{2} \) |
| 73 | \( 1 - 7.43iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.512787714216655186918144921566, −7.59799665112883012176238908362, −7.11700092486704720341781212240, −5.99718122443588471503041715719, −5.54736819971110157387471803479, −4.40820661806549635093945851667, −3.85580457171067128681000282114, −2.79329526303071636073147729426, −2.06223053857462863656775544231, −0.845334557857084165091273945994,
0.929199775790471585841030766669, 2.01973992642498254274299403570, 3.10106050612038777588128712302, 3.59904210729487878928621609585, 4.59246649147157768948892699165, 5.45229599769781043765429822450, 6.17162023581096134142273801745, 7.24639858621517235144009797259, 7.57253881433969621713679757349, 8.314903757243952090094891957963
