L(s) = 1 | − 3.88·5-s + (−2.62 + 0.339i)7-s + 1.48·11-s + 2·13-s + 3.76i·17-s − 0.679i·19-s + 5.20i·23-s + 10.0·25-s + 7.03i·29-s + 6.42·31-s + (10.1 − 1.32i)35-s − 8.71i·37-s − 3.16i·41-s + 8.20·43-s − 9.88·47-s + ⋯ |
L(s) = 1 | − 1.73·5-s + (−0.991 + 0.128i)7-s + 0.446·11-s + 0.554·13-s + 0.913i·17-s − 0.155i·19-s + 1.08i·23-s + 2.01·25-s + 1.30i·29-s + 1.15·31-s + (1.72 − 0.223i)35-s − 1.43i·37-s − 0.493i·41-s + 1.25·43-s − 1.44·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02178647919\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02178647919\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.62 - 0.339i)T \) |
good | 5 | \( 1 + 3.88T + 5T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.76iT - 17T^{2} \) |
| 19 | \( 1 + 0.679iT - 19T^{2} \) |
| 23 | \( 1 - 5.20iT - 23T^{2} \) |
| 29 | \( 1 - 7.03iT - 29T^{2} \) |
| 31 | \( 1 - 6.42T + 31T^{2} \) |
| 37 | \( 1 + 8.71iT - 37T^{2} \) |
| 41 | \( 1 + 3.16iT - 41T^{2} \) |
| 43 | \( 1 - 8.20T + 43T^{2} \) |
| 47 | \( 1 + 9.88T + 47T^{2} \) |
| 53 | \( 1 + 6.42iT - 53T^{2} \) |
| 59 | \( 1 + 7.76iT - 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 8.81T + 67T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 4.44iT - 79T^{2} \) |
| 83 | \( 1 + 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 - 9.88iT - 97T^{2} \) |
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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
−8.796498695137236545107578536386, −8.100582570307051708259327459117, −7.41063587141254230967440910765, −6.75241104461477051687020954227, −6.04283344151542817904053195886, −5.04156381725785124434605020943, −3.91441093015374427705387410004, −3.72557415795863154847411659594, −2.83340995355514430512361160642, −1.27083299168847145997356186623,
0.008426563678264835924467125543, 0.982930203167179147124353260843, 2.81170078779686062408555210434, 3.30206239418491145113301103494, 4.33101816564393559303695480387, 4.58499503913830490280400565479, 6.10491928057944240126850024439, 6.54354846834221362658807253593, 7.40747497620712218202576729455, 7.963829428947653981060400361417