L(s) = 1 | + (−0.5 − 0.866i)3-s + (2 − 3.46i)5-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)11-s + 5·13-s − 3.99·15-s + (−1 − 1.73i)17-s + (0.5 − 0.866i)19-s + (−2 − 1.73i)21-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s + 0.999·27-s + (−1.5 − 2.59i)31-s + (3 − 5.19i)33-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.894 − 1.54i)5-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.904 + 1.56i)11-s + 1.38·13-s − 1.03·15-s + (−0.242 − 0.420i)17-s + (0.114 − 0.198i)19-s + (−0.436 − 0.377i)21-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s + 0.192·27-s + (−0.269 − 0.466i)31-s + (0.522 − 0.904i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56185 - 1.03891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56185 - 1.03891i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.20789065607745085999444056180, −9.383278426191270884312691650164, −8.674789583912594883853952286120, −7.80320321478297949049321688159, −6.75209088194847315910858229290, −5.72751652967955803344668965815, −4.89501701133826085474348048140, −4.11342685931097453348357995218, −1.79068830119891852991131470636, −1.33439324273820332399494020312,
1.67382444222674153211249546940, 3.10366990908361554845274475302, 3.95606596489332238769216122732, 5.54965122300877069066853638637, 6.13787290740666199938066919657, 6.79419013507660823746712956029, 8.337508659190282906682243849569, 8.846075029466886582687522831994, 10.06861961923170233308283198446, 10.81917810987814219042454139658