L(s) = 1 | + (0.328 − 0.328i)3-s + (1.40 + 1.40i)5-s + i·7-s + 2.78i·9-s + (0.444 + 0.444i)11-s + (2.25 − 2.25i)13-s + 0.919·15-s + 4.85·17-s + (−0.114 + 0.114i)19-s + (0.328 + 0.328i)21-s − 3.20i·23-s − 1.06i·25-s + (1.89 + 1.89i)27-s + (−0.997 + 0.997i)29-s − 5.34·31-s + ⋯ |
L(s) = 1 | + (0.189 − 0.189i)3-s + (0.626 + 0.626i)5-s + 0.377i·7-s + 0.928i·9-s + (0.133 + 0.133i)11-s + (0.625 − 0.625i)13-s + 0.237·15-s + 1.17·17-s + (−0.0261 + 0.0261i)19-s + (0.0715 + 0.0715i)21-s − 0.668i·23-s − 0.213i·25-s + (0.365 + 0.365i)27-s + (−0.185 + 0.185i)29-s − 0.959·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.179000277\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.179000277\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-0.328 + 0.328i)T - 3iT^{2} \) |
| 5 | \( 1 + (-1.40 - 1.40i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.444 - 0.444i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.25 + 2.25i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + (0.114 - 0.114i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.20iT - 23T^{2} \) |
| 29 | \( 1 + (0.997 - 0.997i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.34T + 31T^{2} \) |
| 37 | \( 1 + (-2.03 - 2.03i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.57iT - 41T^{2} \) |
| 43 | \( 1 + (-6.86 - 6.86i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 + (-7.64 - 7.64i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.50 - 1.50i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.74 - 1.74i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.96 + 8.96i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.18iT - 71T^{2} \) |
| 73 | \( 1 - 9.04iT - 73T^{2} \) |
| 79 | \( 1 + 9.58T + 79T^{2} \) |
| 83 | \( 1 + (-5.30 + 5.30i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.49iT - 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−9.519140183156877835794242398699, −8.438849622594413974393572094323, −7.916687808555734499389967896042, −7.04108376635593104699928222212, −6.09467599389800423888629797972, −5.53832574967276931320170060468, −4.51034138851591137079982841422, −3.20943135684346717554452838749, −2.50387225840185115344465386588, −1.38157721466672793501096006109,
0.881047365407593488986516423929, 1.92757025621588143934553546310, 3.48424767147224818918438105396, 3.94024916718394690867217243193, 5.24092947939933507246761286479, 5.82777747415740250745667979466, 6.76528268229125943367805632751, 7.58186172406923413892836973288, 8.614870366957241757170812782032, 9.227314001095346341016373651415