L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.382 + 0.923i)3-s + (0.499 − 0.866i)4-s + (−0.130 − 0.991i)6-s + 0.999i·8-s + (−0.707 − 0.707i)9-s − 0.517i·11-s + (0.608 + 0.793i)12-s + (−0.5 − 0.866i)16-s + (0.130 + 0.226i)17-s + (0.965 + 0.258i)18-s + (−1.05 − 0.608i)19-s + (0.258 + 0.448i)22-s + (−0.923 − 0.382i)24-s + 25-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.382 + 0.923i)3-s + (0.499 − 0.866i)4-s + (−0.130 − 0.991i)6-s + 0.999i·8-s + (−0.707 − 0.707i)9-s − 0.517i·11-s + (0.608 + 0.793i)12-s + (−0.5 − 0.866i)16-s + (0.130 + 0.226i)17-s + (0.965 + 0.258i)18-s + (−1.05 − 0.608i)19-s + (0.258 + 0.448i)22-s + (−0.923 − 0.382i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6721869528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6721869528\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.517iT - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1.05 + 0.608i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.991 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.37 + 0.793i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 0.991i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.024013830343133976192402760497, −8.275010000593919438391251137935, −7.45555154669993713340070873812, −6.42714106976649663549093580958, −6.08807799339822934423484238802, −5.11496297983157948944836432772, −4.48548126890727522324084869170, −3.33959257166698745708405786461, −2.31265638023995680367243360431, −0.75490932382727702608727315719,
0.901143901711302470200945087408, 1.99196252910902377694082116764, 2.66735393343724297261621220472, 3.83011876124829947147156682526, 4.84592449073699622343235367249, 5.94384820807946147274887538469, 6.64684237384036550414239702421, 7.28604509953972472383491995014, 7.954759923644099012557573990978, 8.571860536441435720503962749584