| L(s) = 1 | + (−1.88 − 0.663i)2-s + (3.12 + 2.50i)4-s + 0.647i·5-s − 4.63i·7-s + (−4.22 − 6.79i)8-s + (0.429 − 1.22i)10-s − 14.8·11-s − 22.0i·13-s + (−3.07 + 8.73i)14-s + (3.46 + 15.6i)16-s + 18.9i·17-s + (1.76 + 18.9i)19-s + (−1.61 + 2.01i)20-s + (28.0 + 9.87i)22-s + 13.7·23-s + ⋯ |
| L(s) = 1 | + (−0.943 − 0.331i)2-s + (0.780 + 0.625i)4-s + 0.129i·5-s − 0.661i·7-s + (−0.528 − 0.849i)8-s + (0.0429 − 0.122i)10-s − 1.35·11-s − 1.69i·13-s + (−0.219 + 0.624i)14-s + (0.216 + 0.976i)16-s + 1.11i·17-s + (0.0928 + 0.995i)19-s + (−0.0809 + 0.100i)20-s + (1.27 + 0.448i)22-s + 0.596·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3166481329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3166481329\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.88 + 0.663i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.76 - 18.9i)T \) |
| good | 5 | \( 1 - 0.647iT - 25T^{2} \) |
| 7 | \( 1 + 4.63iT - 49T^{2} \) |
| 11 | \( 1 + 14.8T + 121T^{2} \) |
| 13 | \( 1 + 22.0iT - 169T^{2} \) |
| 17 | \( 1 - 18.9iT - 289T^{2} \) |
| 23 | \( 1 - 13.7T + 529T^{2} \) |
| 29 | \( 1 + 21.8T + 841T^{2} \) |
| 31 | \( 1 + 19.3T + 961T^{2} \) |
| 37 | \( 1 - 31.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 38.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 32.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 75.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 113. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 25.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 67.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 92.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 120.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 140. iT - 9.40e3T^{2} \) |
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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
−10.50698447728420142993955512237, −10.00838428422325334598635398998, −8.710957528338234278197573063576, −7.921477134220985877960386253594, −7.46896194861388935831346159357, −6.24082280140292083135388039330, −5.20610808544668626188692552405, −3.62243184023537600571400988617, −2.78998247983125016796121133679, −1.27678033821042419561225771445,
0.15826844873780730405706220928, 1.93214943086181240429288854614, 2.89632010348857039483146904923, 4.83525626898826979517949384125, 5.50294963247153435717391520422, 6.82217858384424803814135814550, 7.28527758042851174535228949312, 8.489227447029658014597251053041, 9.110183065667905603387965527769, 9.719542653707296493826359377261
