L(s) = 1 | + (−2.5 − 4.33i)5-s + (5.24 − 9.07i)7-s + (16.6 − 28.8i)11-s + (11.9 + 20.7i)13-s + 72.5·17-s − 45.9·19-s + (−81.5 − 141. i)23-s + (−12.5 + 21.6i)25-s + (−15.4 + 26.8i)29-s + (−152. − 263. i)31-s − 52.4·35-s + 150.·37-s + (204. + 354. i)41-s + (124. − 215. i)43-s + (77.9 − 135. i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (0.282 − 0.490i)7-s + (0.456 − 0.791i)11-s + (0.255 + 0.442i)13-s + 1.03·17-s − 0.555·19-s + (−0.739 − 1.28i)23-s + (−0.100 + 0.173i)25-s + (−0.0991 + 0.171i)29-s + (−0.882 − 1.52i)31-s − 0.253·35-s + 0.667·37-s + (0.779 + 1.34i)41-s + (0.441 − 0.765i)43-s + (0.241 − 0.419i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.607818322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.607818322\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 7 | \( 1 + (-5.24 + 9.07i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-16.6 + 28.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-11.9 - 20.7i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 72.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + (81.5 + 141. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (15.4 - 26.8i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (152. + 263. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-204. - 354. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-124. + 215. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-77.9 + 135. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-122. - 212. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-233. + 403. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-88.5 - 153. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 45.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 949.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (248. - 430. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (177. - 307. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-204. + 354. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−8.648203221739781962605401204216, −8.036922298519879356694526219313, −7.23838781672640310164571527608, −6.21246199620451052460630690013, −5.58370811668812284453912325661, −4.29413465633777271717149298113, −3.91801705603761851712221063819, −2.59601750005731361265944340294, −1.32350120491914225093296143229, −0.37523058190476067226631585658,
1.24719686508712268388176475774, 2.28001886148146404369786078353, 3.41243320115388669526305473530, 4.20314139752482058207984243531, 5.36797075490827791990115202471, 5.95636940374330251113057149964, 7.08786263095634156686171048672, 7.63335638261460665153123671532, 8.516658477849726813577551782060, 9.320099387657205114901451676359