| L(s) = 1 | + i·2-s − 4-s − 1.73·5-s − i·8-s − 1.73i·10-s + 16-s + 3.46·17-s + 6.92i·19-s + 1.73·20-s − 6i·23-s − 2.00·25-s − 9i·29-s − 3.46i·31-s + i·32-s + 3.46i·34-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.774·5-s − 0.353i·8-s − 0.547i·10-s + 0.250·16-s + 0.840·17-s + 1.58i·19-s + 0.387·20-s − 1.25i·23-s − 0.400·25-s − 1.67i·29-s − 0.622i·31-s + 0.176i·32-s + 0.594i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309567786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309567786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−8.645137568168669793685405913932, −7.996504669831453163445777672603, −7.70198033240149496973681909924, −6.63271677824487500417025567588, −5.97583118186540710589804657705, −5.16421543971665505428944855316, −4.10140459745278632536157767856, −3.66782316406694439840228095954, −2.28855132632752118651242972801, −0.71391549139096931512885407112,
0.72160017936998544412779025253, 1.95847137352763375268975044676, 3.23635022536193382397214840423, 3.63357576461052792250716279637, 4.84092697095315774031840891271, 5.30455612163189332285704815784, 6.58060390273343197066220002550, 7.35763825351725184649023249843, 8.055691217062948911816593381614, 8.867283790315688762138609045961
