L(s) = 1 | − 3i·3-s + 12.5·5-s + (−18.2 − 2.88i)7-s − 9·9-s − 55.4·11-s + 92.1·13-s − 37.5i·15-s + 118. i·17-s − 155. i·19-s + (−8.66 + 54.8i)21-s + 125. i·23-s + 31.6·25-s + 27i·27-s + 131. i·29-s + 66.0·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.11·5-s + (−0.987 − 0.155i)7-s − 0.333·9-s − 1.51·11-s + 1.96·13-s − 0.646i·15-s + 1.68i·17-s − 1.87i·19-s + (−0.0900 + 0.570i)21-s + 1.13i·23-s + 0.253·25-s + 0.192i·27-s + 0.842i·29-s + 0.382·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.216258767\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.216258767\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (18.2 + 2.88i)T \) |
good | 5 | \( 1 - 12.5T + 125T^{2} \) |
| 11 | \( 1 + 55.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 155. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 125. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 131. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 66.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 20.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 355.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 79.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 580. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 496.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 232. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 551. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 437. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 191. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 93.7iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 758. iT - 9.12e5T^{2} \) |
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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
−9.076039931346158572819126173205, −8.518052719842867133278329535635, −7.48388467477934549739554160299, −6.55712249340104884566251668553, −5.94547258871327465119035211592, −5.35531836889166564674811301865, −3.82851574649787662609294687480, −2.87414356700936413389106430082, −1.89957352341158613670973621929, −0.75785331645646641271960170748,
0.71992919545207207091354964862, 2.28229731918361741728567071751, 3.06635917717061385445463986867, 4.08939673590923475460355314909, 5.35102281301701094872073531924, 5.89351905530505041614460209526, 6.52075494628358271858184820322, 7.84122663269544124071937240036, 8.612091182378310763822919043383, 9.478787978632632122919258326187