| L(s) = 1 | + (−0.829 + 1.52i)3-s + (−0.399 + 2.26i)5-s + (−0.715 + 0.600i)7-s + (−1.62 − 2.52i)9-s + (0.843 + 4.78i)11-s + (5.71 − 2.07i)13-s + (−3.11 − 2.48i)15-s + (1.42 − 2.47i)17-s + (−2.50 − 4.34i)19-s + (−0.319 − 1.58i)21-s + (−1.51 − 1.27i)23-s + (−0.277 − 0.101i)25-s + (5.18 − 0.375i)27-s + (1.76 + 0.642i)29-s + (1.02 + 0.856i)31-s + ⋯ |
| L(s) = 1 | + (−0.478 + 0.877i)3-s + (−0.178 + 1.01i)5-s + (−0.270 + 0.226i)7-s + (−0.541 − 0.840i)9-s + (0.254 + 1.44i)11-s + (1.58 − 0.576i)13-s + (−0.804 − 0.642i)15-s + (0.346 − 0.600i)17-s + (−0.575 − 0.996i)19-s + (−0.0696 − 0.345i)21-s + (−0.316 − 0.265i)23-s + (−0.0555 − 0.0202i)25-s + (0.997 − 0.0722i)27-s + (0.328 + 0.119i)29-s + (0.183 + 0.153i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0440 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0440 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.622180 + 0.595376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.622180 + 0.595376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.829 - 1.52i)T \) |
| good | 5 | \( 1 + (0.399 - 2.26i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.715 - 0.600i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.843 - 4.78i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-5.71 + 2.07i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 2.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 + 4.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.51 + 1.27i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.76 - 0.642i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.02 - 0.856i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.00 + 5.20i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (10.5 - 3.82i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.31 - 7.45i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.37 + 7.02i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 2.60T + 53T^{2} \) |
| 59 | \( 1 + (-0.763 + 4.33i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (8.46 - 7.10i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.726 + 0.264i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.62 + 8.01i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.221 - 0.383i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.92 + 1.79i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (14.8 + 5.41i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (7.58 + 13.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.21 + 6.86i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−14.20650362316731931579067476195, −12.80725302579514929854261523230, −11.62798112482015658150670096166, −10.75936250008835115472335376818, −9.939702403276520484825978798046, −8.770941629364597160758249425893, −7.07490974897935293549539659326, −6.05295399109376336921265802535, −4.49757050162436731775435176434, −3.09859084501712322180062277124,
1.21614884681356746460598447036, 3.83533566438019641885723503832, 5.65995253432977415109623587940, 6.47783501653149332169515505196, 8.209770395608804944975749680589, 8.693870545704793973188236207144, 10.53538544694352964930334708174, 11.55129926106719282348693990251, 12.43592141928775035647383130670, 13.43272783254934563573441706973
