| L(s) = 1 | + (−1 − 2i)5-s + 3·9-s − 4i·13-s − 8i·17-s + (−3 + 4i)25-s + 10·29-s + 12i·37-s − 10·41-s + (−3 − 6i)45-s + 7·49-s + 4i·53-s − 10·61-s + (−8 + 4i)65-s + 16i·73-s + 9·81-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s + 9-s − 1.10i·13-s − 1.94i·17-s + (−0.600 + 0.800i)25-s + 1.85·29-s + 1.97i·37-s − 1.56·41-s + (−0.447 − 0.894i)45-s + 49-s + 0.549i·53-s − 1.28·61-s + (−0.992 + 0.496i)65-s + 1.87i·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05471 - 0.651851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05471 - 0.651851i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 12iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−11.74444734168646895307638647682, −10.39603958736884734475203446268, −9.662897648802038270966265082053, −8.583811000670341614167614238718, −7.71380422747132694341431362028, −6.75582600729857951182517827055, −5.20800393949219296852719967035, −4.52579293529095231292682208738, −3.03186958667187543938212924882, −0.989944096178904175251613966692,
1.92886127396355264597428082082, 3.62020807869565141328442857275, 4.47609537034312021662712926526, 6.20057686725751914499571541580, 6.91429802711173902600523257175, 7.900066297661956208511967360682, 8.964854998704914846361387367472, 10.24746998899160334246831870530, 10.66335378443347160423268801462, 11.84408354461813868935945368764
