L(s) = 1 | − 0.732·3-s + 2.73i·5-s + (2 − 1.73i)7-s − 2.46·9-s + 1.46i·11-s − 1.26i·13-s − 2i·15-s − 4i·17-s + 4.73·19-s + (−1.46 + 1.26i)21-s − 1.46i·23-s − 2.46·25-s + 4·27-s + 6.92·29-s + 6.92·31-s + ⋯ |
L(s) = 1 | − 0.422·3-s + 1.22i·5-s + (0.755 − 0.654i)7-s − 0.821·9-s + 0.441i·11-s − 0.351i·13-s − 0.516i·15-s − 0.970i·17-s + 1.08·19-s + (−0.319 + 0.276i)21-s − 0.305i·23-s − 0.492·25-s + 0.769·27-s + 1.28·29-s + 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.517166138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517166138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 - 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 + 1.26iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 9.46iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 5.66iT - 61T^{2} \) |
| 67 | \( 1 - 9.46iT - 67T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 - 1.07iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.634611055441581144336654905634, −8.331296026421885016985619968383, −7.79424689992802731710705067168, −6.85165027903670139252040360106, −6.39817283727256316719675304594, −5.19964954310748762484086287959, −4.62132978223779372232419129059, −3.22002225352805589147018759302, −2.64157986983474498769625291627, −0.996521312305338989560089000490,
0.78831150340855633856001371166, 1.93689706775363467564271433827, 3.24895834688436860532750468647, 4.47728641549432196620877133598, 5.25055608999607256001843169193, 5.68269782999461804617552959280, 6.65127397876537475189678540714, 7.902367443474173851548421253099, 8.635563529335969910465552003616, 8.797386924713586596707751590123