L(s) = 1 | + (−0.292 − 0.292i)2-s + (2.84 − 1.17i)3-s − 1.82i·4-s + (0.0154 + 0.0374i)5-s + (−1.17 − 0.488i)6-s + (−0.341 + 0.825i)7-s + (−1.12 + 1.12i)8-s + (4.58 − 4.58i)9-s + (0.00641 − 0.0154i)10-s + (2.69 + 1.11i)11-s + (−2.15 − 5.20i)12-s − 4.69i·13-s + (0.341 − 0.141i)14-s + (0.0881 + 0.0881i)15-s − 3.00·16-s + (−0.147 + 4.12i)17-s + ⋯ |
L(s) = 1 | + (−0.207 − 0.207i)2-s + (1.64 − 0.680i)3-s − 0.914i·4-s + (0.00693 + 0.0167i)5-s + (−0.480 − 0.199i)6-s + (−0.129 + 0.311i)7-s + (−0.396 + 0.396i)8-s + (1.52 − 1.52i)9-s + (0.00202 − 0.00490i)10-s + (0.813 + 0.337i)11-s + (−0.621 − 1.50i)12-s − 1.30i·13-s + (0.0913 − 0.0378i)14-s + (0.0227 + 0.0227i)15-s − 0.750·16-s + (−0.0357 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0822 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0822 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59673 - 1.73386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59673 - 1.73386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (0.147 - 4.12i)T \) |
| 43 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (0.292 + 0.292i)T + 2iT^{2} \) |
| 3 | \( 1 + (-2.84 + 1.17i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.0154 - 0.0374i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (0.341 - 0.825i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.69 - 1.11i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + 4.69iT - 13T^{2} \) |
| 19 | \( 1 + (-1.40 - 1.40i)T + 19iT^{2} \) |
| 23 | \( 1 + (7.48 + 3.09i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.652 - 1.57i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-6.85 + 2.83i)T + (21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (8.16 - 3.38i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.849 - 2.05i)T + (-28.9 - 28.9i)T^{2} \) |
| 47 | \( 1 - 13.0iT - 47T^{2} \) |
| 53 | \( 1 + (9.03 + 9.03i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.17 - 1.17i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.00 - 12.0i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 + (-12.1 + 5.03i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (3.38 + 8.18i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-7.47 - 3.09i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-8.05 - 8.05i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.27iT - 89T^{2} \) |
| 97 | \( 1 + (-1.77 - 4.27i)T + (-68.5 + 68.5i)T^{2} \) |
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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
−9.986586968517400047398577786708, −9.287376780687851265961902869764, −8.382687775174237765599119130908, −7.951456424055028217698659751531, −6.63376004191565754922333768927, −6.00850024131547862743719016095, −4.46190445275558983718504630017, −3.22804450030838594954118313512, −2.24808669869308820681242377828, −1.22197799005775682044354921978,
2.06182942417865690306677522901, 3.28069104540291649002155699466, 3.84805545602058024706225619220, 4.75820680877158746482088759079, 6.66214783742284934118963876724, 7.34430448555359941904697058227, 8.200191184773425318095224740286, 9.049696734466100719537356792031, 9.294995220221859520821927022878, 10.24569592298241865793297935134