| L(s) = 1 | + 3.13i·3-s + i·5-s − 7-s − 6.82·9-s − 4.07i·11-s − 2.86i·13-s − 3.13·15-s + 7.69·17-s − 2.79i·19-s − 3.13i·21-s − 7.79·23-s − 25-s − 11.9i·27-s − 9.95i·29-s + 0.232·31-s + ⋯ |
| L(s) = 1 | + 1.80i·3-s + 0.447i·5-s − 0.377·7-s − 2.27·9-s − 1.22i·11-s − 0.795i·13-s − 0.809·15-s + 1.86·17-s − 0.640i·19-s − 0.684i·21-s − 1.62·23-s − 0.200·25-s − 2.30i·27-s − 1.84i·29-s + 0.0417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.017710296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017710296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.13iT - 3T^{2} \) |
| 11 | \( 1 + 4.07iT - 11T^{2} \) |
| 13 | \( 1 + 2.86iT - 13T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 + 2.79iT - 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 + 9.95iT - 29T^{2} \) |
| 31 | \( 1 - 0.232T + 31T^{2} \) |
| 37 | \( 1 + 4.98iT - 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 - 9.54iT - 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + 3.85iT - 53T^{2} \) |
| 59 | \( 1 + 7.86iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 + 11.5iT - 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 + 0.872T + 79T^{2} \) |
| 83 | \( 1 - 8.33iT - 83T^{2} \) |
| 89 | \( 1 + 5.79T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.321565070056558289689089059103, −8.175300950252591615040148584577, −7.88618441806688516871135896087, −6.23023291971282205520062982297, −5.79781331084613108239687406261, −5.03643752592554550495885569779, −3.89279258193799724077196460836, −3.42464604382225781101796366519, −2.65725259967567174894193315628, −0.36327477123368456835839083997,
1.29018874392639057365076948815, 1.85159591907131527854231529278, 3.01500696032895648231075912710, 4.16488515342199814797058418450, 5.46996087341438545585986980055, 5.96384167833292029326532245250, 7.11961057507601593813090770403, 7.22919789985984740827549666799, 8.182717032130772740950012693631, 8.792724224303467133623457094429
