| L(s) = 1 | + (−1.39 + 0.245i)2-s + (−2.97 + 0.418i)3-s + (1.87 − 0.684i)4-s + (5.54 − 6.61i)5-s + (4.03 − 1.31i)6-s + (7.83 + 2.85i)7-s + (−2.44 + 1.41i)8-s + (8.64 − 2.48i)9-s + (−6.10 + 10.5i)10-s + (−10.8 − 12.8i)11-s + (−5.29 + 2.81i)12-s + (0.524 − 2.97i)13-s + (−11.6 − 2.04i)14-s + (−13.7 + 21.9i)15-s + (3.06 − 2.57i)16-s + (8.43 + 4.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.990 + 0.139i)3-s + (0.469 − 0.171i)4-s + (1.10 − 1.32i)5-s + (0.672 − 0.218i)6-s + (1.11 + 0.407i)7-s + (−0.306 + 0.176i)8-s + (0.961 − 0.276i)9-s + (−0.610 + 1.05i)10-s + (−0.983 − 1.17i)11-s + (−0.441 + 0.234i)12-s + (0.0403 − 0.229i)13-s + (−0.829 − 0.146i)14-s + (−0.914 + 1.46i)15-s + (0.191 − 0.160i)16-s + (0.496 + 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.770323 - 0.225348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.770323 - 0.225348i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.245i)T \) |
| 3 | \( 1 + (2.97 - 0.418i)T \) |
| good | 5 | \( 1 + (-5.54 + 6.61i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-7.83 - 2.85i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (10.8 + 12.8i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (-0.524 + 2.97i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-8.43 - 4.87i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.84 - 6.66i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-10.1 - 27.8i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (10.5 - 1.86i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (10.8 - 3.93i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (11.5 - 20.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (16.7 + 2.94i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-18.0 + 15.1i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (5.67 - 15.6i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 75.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (38.6 - 46.1i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-32.1 - 11.6i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-6.53 + 37.0i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-44.0 - 25.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-49.2 - 85.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.0 + 79.9i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (15.8 - 2.80i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-14.0 + 8.09i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (101. - 85.2i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−15.44951406976819285820924153438, −13.74015622142948080689533196053, −12.59517926853290457650218297148, −11.39603090768557399976781996781, −10.31201103143598750384660982408, −9.075617442566873236751535468153, −7.939409860290884159421490989189, −5.73857685115579881784901882139, −5.23889869829205570672871380794, −1.34437959257986767643153912992,
2.07724977452773776644285081210, 5.13289823889981459456228717844, 6.70343647371311202658993173119, 7.59975714653899447385896847908, 9.800009875912268214626635320784, 10.57938475840893397492130989648, 11.30329607011693468790171371156, 12.80230830303873372596826820238, 14.20748571995648656676224274125, 15.23773494361804746413330517413
