L(s) = 1 | + (2 + i)5-s + i·7-s − 3·11-s − i·13-s + 5i·17-s + 8·19-s + 2i·23-s + (3 + 4i)25-s − 29-s − 2·31-s + (−1 + 2i)35-s + 10i·37-s + 6·41-s + 4i·43-s − 11i·47-s + ⋯ |
L(s) = 1 | + (0.894 + 0.447i)5-s + 0.377i·7-s − 0.904·11-s − 0.277i·13-s + 1.21i·17-s + 1.83·19-s + 0.417i·23-s + (0.600 + 0.800i)25-s − 0.185·29-s − 0.359·31-s + (−0.169 + 0.338i)35-s + 1.64i·37-s + 0.937·41-s + 0.609i·43-s − 1.60i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.807994043\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.807994043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 11iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−9.855449712044065944039572953730, −9.185605115254482351151244510556, −8.132438033440835325101428994858, −7.44557662543581336673537201119, −6.38805764852827934882428425379, −5.62962321831618817269077185998, −5.00884605097615267223011452522, −3.48356144171216555923539033915, −2.65743497227812755452696957885, −1.47450250584823845651620228032,
0.803691589757017357547531986139, 2.20855245488941755405519631860, 3.21751479034159165091794271975, 4.61246984879818660047458870606, 5.30184409228837645111288328745, 6.06019852018519156107118067585, 7.26746247473276944102929865160, 7.72598046010930576964985762851, 9.035703162138487266246226417885, 9.435936388611970654287193969951