L(s) = 1 | − 1.55i·2-s − 0.410·4-s + 2.64i·7-s − 2.46i·8-s + 3·9-s + 4.10·14-s − 4.65·16-s − 4.65i·18-s + 9.58·23-s + 5·25-s − 1.08i·28-s − 1.36i·29-s + 2.28i·32-s − 1.23·36-s − 1.36·37-s + ⋯ |
L(s) = 1 | − 1.09i·2-s − 0.205·4-s + 0.999i·7-s − 0.872i·8-s + 9-s + 1.09·14-s − 1.16·16-s − 1.09i·18-s + 1.99·23-s + 25-s − 0.205i·28-s − 0.254i·29-s + 0.404i·32-s − 0.205·36-s − 0.224·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46793 - 1.17490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46793 - 1.17490i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - 2.64iT \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.55iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9.58T + 23T^{2} \) |
| 29 | \( 1 + 1.36iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 9.77iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 16.3T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 12.5iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.23138234037296314805678441655, −9.261899665770175275052651086030, −8.730756919216017161665233347865, −7.29564626733532112423889578107, −6.69234476205945857843901847432, −5.42143230642945611420140037140, −4.41467805800734792580816699415, −3.25869973863904166559423853896, −2.35771103081175849455557496045, −1.17914508720037327233992850196,
1.29532816596362424247493935780, 3.00541581884440210593320067391, 4.39296488393907645838339870708, 5.07637617109210054158165553686, 6.30335786446246957658498274024, 7.14126345505036444453395935557, 7.38096293495672983280972882409, 8.506720109651622766680754262274, 9.346108207206937836699937954543, 10.47308528724379531552123728660