| L(s) = 1 | − 2i·3-s − i·5-s − 2·7-s − 9-s + 4i·11-s + 6i·13-s − 2·15-s + 2·17-s + 8i·19-s + 4i·21-s − 6·23-s − 25-s − 4i·27-s + 2i·29-s − 4·31-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.447i·5-s − 0.755·7-s − 0.333·9-s + 1.20i·11-s + 1.66i·13-s − 0.516·15-s + 0.485·17-s + 1.83i·19-s + 0.872i·21-s − 1.25·23-s − 0.200·25-s − 0.769i·27-s + 0.371i·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.064744581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064744581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.700221147795872853924297097288, −9.022025050003019166264077043445, −7.83836494598430177060851685915, −7.42199661117713160895476820505, −6.46145017983366403909347136228, −5.94549874758696830472340772572, −4.55162433626430343127034466820, −3.74526964187479998232313419919, −2.12190190644461181470731397590, −1.48696706425844324269749263446,
0.44771859190856424942134289020, 2.77751170045628542749338882694, 3.35973553900055267476115977278, 4.26940098367687164259175307015, 5.46976262182893041594889029283, 5.97029938653171017300906150906, 7.12827186216962209056827694429, 8.028615783301155763290298911097, 8.981288937050764817614876360543, 9.649125828234924965367259372500
