| L(s) = 1 | − 1.71i·3-s − i·5-s + 2.02·7-s + 0.0697·9-s − 4.96i·11-s + 2.57i·13-s − 1.71·15-s + 7.49·17-s − i·19-s − 3.46i·21-s − 2.35·23-s − 25-s − 5.25i·27-s − 4.27i·29-s − 0.942·31-s + ⋯ |
| L(s) = 1 | − 0.988i·3-s − 0.447i·5-s + 0.766·7-s + 0.0232·9-s − 1.49i·11-s + 0.712i·13-s − 0.441·15-s + 1.81·17-s − 0.229i·19-s − 0.757i·21-s − 0.490·23-s − 0.200·25-s − 1.01i·27-s − 0.793i·29-s − 0.169·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236555281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236555281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.71iT - 3T^{2} \) |
| 7 | \( 1 - 2.02T + 7T^{2} \) |
| 11 | \( 1 + 4.96iT - 11T^{2} \) |
| 13 | \( 1 - 2.57iT - 13T^{2} \) |
| 17 | \( 1 - 7.49T + 17T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 + 4.27iT - 29T^{2} \) |
| 31 | \( 1 + 0.942T + 31T^{2} \) |
| 37 | \( 1 - 6.80iT - 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 + 0.842iT - 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 4.76iT - 53T^{2} \) |
| 59 | \( 1 - 2.77iT - 59T^{2} \) |
| 61 | \( 1 - 2.46iT - 61T^{2} \) |
| 67 | \( 1 + 5.12iT - 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 4.50T + 79T^{2} \) |
| 83 | \( 1 + 8.77iT - 83T^{2} \) |
| 89 | \( 1 + 0.178T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.168547842612060965230862872304, −7.913006930059928492211526440601, −7.11095346751833749392279456098, −6.09260232743576301259348276175, −5.66807730463331133501376487084, −4.61203503802309700632607772600, −3.72547596810493675742744179744, −2.60083783938663412686798824188, −1.47324854975775740640917870164, −0.806828364774522701956059716735,
1.34205933310670619326397788922, 2.46588827108240895596600521812, 3.60932629652211010926807516548, 4.19009092360566309823883207979, 5.16076670835367646250373566315, 5.55233850780488959268874156054, 6.81934089983473741442170144952, 7.64099513784331575541289682719, 7.971318801003839469691758880686, 9.259681902619083151426077316716
