| L(s) = 1 | − 3-s − 3i·5-s − 4i·7-s + 9-s + 4i·11-s + 3i·15-s − 3·17-s + 4i·19-s + 4i·21-s + 8·23-s − 4·25-s − 27-s − 5·29-s + 8i·31-s − 4i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34i·5-s − 1.51i·7-s + 0.333·9-s + 1.20i·11-s + 0.774i·15-s − 0.727·17-s + 0.917i·19-s + 0.872i·21-s + 1.66·23-s − 0.800·25-s − 0.192·27-s − 0.928·29-s + 1.43i·31-s − 0.696i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7847927793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7847927793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 9iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.522525826896377967531215381340, −7.76637048939331877004780187334, −7.00596073447895627894797989208, −6.55372862008463602548803304949, −5.28257615153433892031959312489, −4.75482346141716945675330516069, −4.31203668222134247431411608120, −3.31424841318926654391223287153, −1.62471999541299226337073897909, −1.09525821011471431421559743508,
0.26793771357330835676967309208, 2.04435212516069939034141087746, 2.80709344161344085548204362283, 3.45963494330287419390559883677, 4.69617670571205243859206081177, 5.63595619787109142095159075710, 5.99327589740384148546722784117, 6.80792002454392283805649812523, 7.35595226247457575077382545578, 8.440943786592680144133768563004
