| L(s) = 1 | + (−0.495 + 0.205i)2-s + (1.69 + 0.337i)3-s + (−2.62 + 2.62i)4-s + (3.55 + 5.32i)5-s + (−0.910 + 0.181i)6-s + (3.48 − 5.21i)7-s + (1.58 − 3.81i)8-s + (2.77 + 1.14i)9-s + (−2.85 − 1.90i)10-s + (2.24 + 11.2i)11-s + (−5.34 + 3.57i)12-s + (−16.7 − 16.7i)13-s + (−0.656 + 3.29i)14-s + (4.24 + 10.2i)15-s − 12.6i·16-s + (5.36 − 16.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.247 + 0.102i)2-s + (0.566 + 0.112i)3-s + (−0.656 + 0.656i)4-s + (0.711 + 1.06i)5-s + (−0.151 + 0.0301i)6-s + (0.498 − 0.745i)7-s + (0.197 − 0.477i)8-s + (0.307 + 0.127i)9-s + (−0.285 − 0.190i)10-s + (0.203 + 1.02i)11-s + (−0.445 + 0.297i)12-s + (−1.28 − 1.28i)13-s + (−0.0468 + 0.235i)14-s + (0.283 + 0.683i)15-s − 0.789i·16-s + (0.315 − 0.948i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05853 + 0.477128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05853 + 0.477128i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 - 0.337i)T \) |
| 17 | \( 1 + (-5.36 + 16.1i)T \) |
| good | 2 | \( 1 + (0.495 - 0.205i)T + (2.82 - 2.82i)T^{2} \) |
| 5 | \( 1 + (-3.55 - 5.32i)T + (-9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-3.48 + 5.21i)T + (-18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-2.24 - 11.2i)T + (-111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (16.7 + 16.7i)T + 169iT^{2} \) |
| 19 | \( 1 + (-6.69 + 2.77i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (17.8 - 3.54i)T + (488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-24.4 + 16.3i)T + (321. - 776. i)T^{2} \) |
| 31 | \( 1 + (3.99 - 20.0i)T + (-887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-49.2 - 9.79i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (17.1 - 25.5i)T + (-643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (57.3 + 23.7i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (23.0 + 23.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (7.98 - 3.30i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (21.0 - 50.8i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (13.8 + 9.26i)T + (1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 1.37iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (80.9 + 16.1i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (63.6 + 95.3i)T + (-2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-6.44 - 32.3i)T + (-5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-52.5 - 126. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (95.1 - 95.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-46.1 + 30.8i)T + (3.60e3 - 8.69e3i)T^{2} \) |
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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
−15.13318357375880554366723136276, −14.24787818910908589401615164372, −13.44614090914517546108018792946, −12.10024834636406124261948567183, −10.19680155072082592797860336812, −9.726075261411245185616725634162, −7.901189168046367166096562461575, −7.12379544536385792150176877936, −4.74135402767965351947710111696, −2.90050922315664858374775292333,
1.73955987995647410233090726070, 4.66800520735461303687253949732, 5.91381654059221999809650829956, 8.294731736657526049135941239223, 9.072947244303156859991545702021, 9.943199977573725756275356089178, 11.74443028332461923932408604464, 13.00745548300451993885493503344, 14.11025719549256372697262892033, 14.72650860477569328480575932130
