| L(s) = 1 | + i·2-s − 4-s + 2i·5-s − 2i·7-s − i·8-s − 2·10-s − 4i·11-s + 2·14-s + 16-s + 6i·19-s − 2i·20-s + 4·22-s − 4·23-s + 25-s + 2i·28-s − 8·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.894i·5-s − 0.755i·7-s − 0.353i·8-s − 0.632·10-s − 1.20i·11-s + 0.534·14-s + 0.250·16-s + 1.37i·19-s − 0.447i·20-s + 0.852·22-s − 0.834·23-s + 0.200·25-s + 0.377i·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328409987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328409987\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 16iT - 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 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
−8.736306040921971567778244045865, −8.130331027313402014022915867052, −7.35509082529192321217101711958, −6.83997034662755629319492260281, −5.88733544450536283160453938632, −5.53352626548846282311455844000, −4.04590050970486042381792713437, −3.70538416450980619881243672750, −2.57990884616682223021691345415, −1.06295113737662403086391330604,
0.47953625002604668325973577242, 1.88266447847373429979879791430, 2.45380824021374992298400825114, 3.74982092607709746520763658419, 4.56898885540164786180075272835, 5.18099701351502225675308264976, 5.95056011307064613638986078545, 7.10307472356972986103340027609, 7.80317539885099716024793577809, 8.848216874534899956306174034671
