| L(s) = 1 | + (2.05 − 0.989i)2-s + (1.99 − 2.49i)4-s + (0.131 − 0.0405i)5-s + (0.934 − 1.61i)7-s + (0.607 − 2.66i)8-s + (0.230 − 0.213i)10-s + (−1.63 − 2.05i)11-s + (1.27 + 1.18i)13-s + (0.318 − 4.24i)14-s + (0.0382 + 0.167i)16-s + (−1.45 − 0.448i)17-s + (4.83 + 0.729i)19-s + (0.160 − 0.409i)20-s + (−5.39 − 2.59i)22-s + (−1.59 + 4.07i)23-s + ⋯ |
| L(s) = 1 | + (1.45 − 0.699i)2-s + (0.996 − 1.24i)4-s + (0.0588 − 0.0181i)5-s + (0.353 − 0.611i)7-s + (0.214 − 0.940i)8-s + (0.0727 − 0.0675i)10-s + (−0.493 − 0.618i)11-s + (0.354 + 0.329i)13-s + (0.0851 − 1.13i)14-s + (0.00956 + 0.0418i)16-s + (−0.352 − 0.108i)17-s + (1.10 + 0.167i)19-s + (0.0359 − 0.0916i)20-s + (−1.14 − 0.553i)22-s + (−0.333 + 0.848i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42522 - 1.63934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42522 - 1.63934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (5.73 - 3.17i)T \) |
| good | 2 | \( 1 + (-2.05 + 0.989i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (-0.131 + 0.0405i)T + (4.13 - 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.934 + 1.61i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.63 + 2.05i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 1.18i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.45 + 0.448i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-4.83 - 0.729i)T + (18.1 + 5.60i)T^{2} \) |
| 23 | \( 1 + (1.59 - 4.07i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.00236 + 0.0315i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (2.33 + 1.59i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (0.00465 + 0.00806i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.49 - 3.61i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-3.50 + 4.39i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-5.67 + 5.26i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (0.483 + 2.11i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (10.0 - 6.84i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-7.65 - 1.15i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (-4.78 - 12.2i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (3.48 + 3.23i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-6.00 + 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.321 + 4.29i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (0.574 + 7.67i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-1.00 - 1.25i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.45170716274424632067388066303, −10.64097516187152718797094284817, −9.662955666701151408155059765640, −8.295117709204114108309938566052, −7.19265765714252943687769354568, −5.90922394574545477153502821711, −5.15471763485030566596456076251, −4.02198703148490004798166643929, −3.16251435139980481843194848190, −1.65190077121812105792423051752,
2.37324796628223408588321727168, 3.67117057040567594369503856784, 4.84761745355647422699952516537, 5.53301551189670960068717787629, 6.51395578766786776239859960043, 7.49463086817591449219859664574, 8.417694941563438020043776498272, 9.691589797156786992093647853416, 10.81812388638702444589576595927, 12.02779113256448514244383052080
