| L(s) = 1 | + (−4.90 + 0.515i)3-s + (−1.45 − 2.51i)5-s + (7.26 + 1.54i)7-s + (15.0 − 3.19i)9-s + (13.9 + 12.5i)11-s + (−0.441 + 0.991i)13-s + (8.43 + 11.6i)15-s + (18.8 − 17.0i)17-s + (−16.2 + 7.22i)19-s + (−36.4 − 3.83i)21-s + (23.7 + 7.72i)23-s + (8.26 − 14.3i)25-s + (−29.8 + 9.69i)27-s + (8.46 − 11.6i)29-s + (−28.1 + 12.9i)31-s + ⋯ |
| L(s) = 1 | + (−1.63 + 0.171i)3-s + (−0.290 − 0.503i)5-s + (1.03 + 0.220i)7-s + (1.66 − 0.354i)9-s + (1.26 + 1.14i)11-s + (−0.0339 + 0.0763i)13-s + (0.562 + 0.774i)15-s + (1.11 − 1.00i)17-s + (−0.853 + 0.380i)19-s + (−1.73 − 0.182i)21-s + (1.03 + 0.336i)23-s + (0.330 − 0.572i)25-s + (−1.10 + 0.359i)27-s + (0.292 − 0.401i)29-s + (−0.908 + 0.418i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.923774 + 0.104737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.923774 + 0.104737i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (28.1 - 12.9i)T \) |
| good | 3 | \( 1 + (4.90 - 0.515i)T + (8.80 - 1.87i)T^{2} \) |
| 5 | \( 1 + (1.45 + 2.51i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-7.26 - 1.54i)T + (44.7 + 19.9i)T^{2} \) |
| 11 | \( 1 + (-13.9 - 12.5i)T + (12.6 + 120. i)T^{2} \) |
| 13 | \( 1 + (0.441 - 0.991i)T + (-113. - 125. i)T^{2} \) |
| 17 | \( 1 + (-18.8 + 17.0i)T + (30.2 - 287. i)T^{2} \) |
| 19 | \( 1 + (16.2 - 7.22i)T + (241. - 268. i)T^{2} \) |
| 23 | \( 1 + (-23.7 - 7.72i)T + (427. + 310. i)T^{2} \) |
| 29 | \( 1 + (-8.46 + 11.6i)T + (-259. - 799. i)T^{2} \) |
| 37 | \( 1 + (-28.9 - 16.7i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (6.05 - 57.6i)T + (-1.64e3 - 349. i)T^{2} \) |
| 43 | \( 1 + (-10.7 - 24.2i)T + (-1.23e3 + 1.37e3i)T^{2} \) |
| 47 | \( 1 + (-50.9 + 37.0i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-4.39 - 20.6i)T + (-2.56e3 + 1.14e3i)T^{2} \) |
| 59 | \( 1 + (1.79 + 17.0i)T + (-3.40e3 + 723. i)T^{2} \) |
| 61 | \( 1 - 52.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (63.7 + 110. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (27.8 - 5.92i)T + (4.60e3 - 2.05e3i)T^{2} \) |
| 73 | \( 1 + (-17.9 - 16.1i)T + (557. + 5.29e3i)T^{2} \) |
| 79 | \( 1 + (-48.6 + 43.7i)T + (652. - 6.20e3i)T^{2} \) |
| 83 | \( 1 + (1.69 + 0.178i)T + (6.73e3 + 1.43e3i)T^{2} \) |
| 89 | \( 1 + (-78.5 + 25.5i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (41.1 + 126. i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−12.70227178081845254166583403528, −11.93447233572680473882941020674, −11.47243576677092218378501535617, −10.28932438611433129466157826927, −9.134041415712060814794845862577, −7.58915934928454120247530453217, −6.40997703248346355743711552855, −5.08364942408874384629959149987, −4.41706703079754897443068433120, −1.23605397769343508830990769093,
1.09026130578875362466503268655, 3.94337188702757742692875297102, 5.36943375531494534587714273543, 6.34754518726972870591536065597, 7.41360609311978140963433300254, 8.839154717451675514810649865352, 10.72061476959829125850922887641, 10.97875960311140472276731482490, 11.86671556103501625843654889164, 12.80724817246010426748487553658
