L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.130 + 0.991i)3-s + (0.499 + 0.866i)4-s + (0.608 − 0.793i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (−1.67 − 0.965i)11-s + (−0.923 + 0.382i)12-s + (−0.5 + 0.866i)16-s + 1.21·17-s + (0.707 + 0.707i)18-s − 0.261i·19-s + (0.965 + 1.67i)22-s + (0.991 + 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.130 + 0.991i)3-s + (0.499 + 0.866i)4-s + (0.608 − 0.793i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (−1.67 − 0.965i)11-s + (−0.923 + 0.382i)12-s + (−0.5 + 0.866i)16-s + 1.21·17-s + (0.707 + 0.707i)18-s − 0.261i·19-s + (0.965 + 1.67i)22-s + (0.991 + 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4618397771\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4618397771\) |
\(L(1)\) |
|
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 | \( 1 + (0.130 - 0.991i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.21T + T^{2} \) |
| 19 | \( 1 + 0.261iT - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.793 + 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + 1.98iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - 1.84T + T^{2} \) |
| 97 | \( 1 + (-1.37 - 0.793i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.778178420353285609838963442110, −7.920985268237436677493130053315, −7.58000151167132674261616882032, −6.27705966406622724941478310659, −5.50787895003418424006173432130, −4.81371631934279557705185845304, −3.42365762332981742516704063107, −3.26918123330825873124953318491, −2.08440939855505136947756484737, −0.38602285949617989978821013844,
1.19120073182921632226156055535, 2.23488287064216846161298092928, 2.94463739807692680315814953332, 4.65963642619755271153690374794, 5.47169111354512640093591025103, 6.01442136393881073589190649356, 6.91626117182716359791313353627, 7.62942233497611769174648030509, 7.923301108870627126293792712334, 8.600187651175577607066687964574