L(s) = 1 | + 1.69i·5-s + (−2.56 + 0.662i)7-s + 3.02i·11-s + 6.04i·13-s − 4.34i·17-s − 1.12·19-s + 3.02i·23-s + 2.12·25-s − 2·29-s + (−1.12 − 4.34i)35-s + 7.12·37-s + 7.73i·41-s + 8.10i·43-s − 10.2·47-s + (6.12 − 3.39i)49-s + ⋯ |
L(s) = 1 | + 0.758i·5-s + (−0.968 + 0.250i)7-s + 0.910i·11-s + 1.67i·13-s − 1.05i·17-s − 0.257·19-s + 0.629i·23-s + 0.424·25-s − 0.371·29-s + (−0.189 − 0.734i)35-s + 1.17·37-s + 1.20i·41-s + 1.23i·43-s − 1.49·47-s + (0.874 − 0.484i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6954242094\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6954242094\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.56 - 0.662i)T \) |
good | 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 - 6.04iT - 13T^{2} \) |
| 17 | \( 1 + 4.34iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 3.02iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 7.73iT - 41T^{2} \) |
| 43 | \( 1 - 8.10iT - 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 9.43iT - 61T^{2} \) |
| 67 | \( 1 + 2.06iT - 67T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.39iT - 73T^{2} \) |
| 79 | \( 1 - 4.71iT - 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 7.73iT - 89T^{2} \) |
| 97 | \( 1 - 8.68iT - 97T^{2} \) |
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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
−9.134185685900352444250081184987, −7.959596249949823288848478333256, −7.21574866776214674924999139767, −6.58841379373378395649792164212, −6.24430031381970045289119395118, −4.96986241335451230661389005502, −4.33817469635138339313694855890, −3.29718623508655407620193637815, −2.61442834826053004758013146829, −1.62636631647181878795950817625,
0.21555920372981591326672627010, 1.10820382952190459478350692553, 2.59114881439290876133384350252, 3.40590503258633021842193901971, 4.10787661978454326185073248864, 5.19050717702498478032606852035, 5.83532392847784939186540283921, 6.41302934211802151731230951815, 7.39760502016339604199494044231, 8.224471848534990998539240713612