L(s) = 1 | + (0.984 − 0.173i)2-s + (0.939 + 0.342i)3-s + (0.939 − 0.342i)4-s + (0.984 + 0.173i)6-s + (0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + 0.999·12-s + (−0.469 − 1.75i)13-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)18-s + (−0.866 − 0.5i)23-s + (0.984 − 0.173i)24-s + (−0.866 + 0.5i)25-s + (−0.766 − 1.64i)26-s + (0.500 + 0.866i)27-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (0.939 + 0.342i)3-s + (0.939 − 0.342i)4-s + (0.984 + 0.173i)6-s + (0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + 0.999·12-s + (−0.469 − 1.75i)13-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)18-s + (−0.866 − 0.5i)23-s + (0.984 − 0.173i)24-s + (−0.866 + 0.5i)25-s + (−0.766 − 1.64i)26-s + (0.500 + 0.866i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.239528421\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.239528421\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (0.469 + 1.75i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (0.424 - 1.58i)T + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - 1.96iT - T^{2} \) |
| 73 | \( 1 - 0.347iT - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.642446320200085397528277941030, −7.948822363616435745929850973834, −7.37852986437837294536612669054, −6.49119190228510645840648586750, −5.44616396470946175846561402695, −4.97732659308861107463163396754, −3.95365922905943485066200967825, −3.26638758463652674263898372494, −2.60166750000868072038282068810, −1.53989556737742986646146481336,
1.86380271041730921038176491567, 2.24552741887518829739683369115, 3.47099454825626821147586218183, 4.12149689740314719522515177164, 4.75856775751856184166617102305, 6.02775674886596770659116358903, 6.51029815622456891295452066289, 7.38145726261307972663337153730, 7.85211201027281116428488820948, 8.681646008468752205070185981408