L(s) = 1 | + (2.05 + 2.05i)3-s + (1.58 + 1.58i)5-s − 10.3·7-s − 0.519i·9-s + (−14.0 + 14.0i)11-s + (−9.77 + 9.77i)13-s + 6.51i·15-s + 7.10·17-s + (16.6 + 16.6i)19-s + (−21.4 − 21.4i)21-s − 24.8·23-s + 5.00i·25-s + (19.6 − 19.6i)27-s + (11.2 − 11.2i)29-s + 7.04i·31-s + ⋯ |
L(s) = 1 | + (0.686 + 0.686i)3-s + (0.316 + 0.316i)5-s − 1.48·7-s − 0.0576i·9-s + (−1.27 + 1.27i)11-s + (−0.751 + 0.751i)13-s + 0.434i·15-s + 0.417·17-s + (0.874 + 0.874i)19-s + (−1.01 − 1.01i)21-s − 1.07·23-s + 0.200i·25-s + (0.726 − 0.726i)27-s + (0.389 − 0.389i)29-s + 0.227i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.333263 + 1.09315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.333263 + 1.09315i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.58 - 1.58i)T \) |
good | 3 | \( 1 + (-2.05 - 2.05i)T + 9iT^{2} \) |
| 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 + (14.0 - 14.0i)T - 121iT^{2} \) |
| 13 | \( 1 + (9.77 - 9.77i)T - 169iT^{2} \) |
| 17 | \( 1 - 7.10T + 289T^{2} \) |
| 19 | \( 1 + (-16.6 - 16.6i)T + 361iT^{2} \) |
| 23 | \( 1 + 24.8T + 529T^{2} \) |
| 29 | \( 1 + (-11.2 + 11.2i)T - 841iT^{2} \) |
| 31 | \( 1 - 7.04iT - 961T^{2} \) |
| 37 | \( 1 + (-11.2 - 11.2i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-37.0 + 37.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 0.176iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (32.3 + 32.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (25.8 - 25.8i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (36.8 - 36.8i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-12.8 - 12.8i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 64.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 19.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 48.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-18.5 - 18.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 43.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115.T + 9.40e3T^{2} \) |
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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
−12.02811638428556674213286280831, −10.31638036631070956005025665997, −9.795101381769692554836701494405, −9.489176570920332078677232484923, −7.992968557775835323856742262701, −7.01756985114569739179042864464, −5.94130713562563246247286724916, −4.56061134728910341705988977536, −3.36695125951056116680495115687, −2.41175524211877586835213381806,
0.46240033261669421673792382693, 2.56518132841063055841028981692, 3.23021190602546380476004571852, 5.21401296904855224591679670669, 6.10444458571768604512947718107, 7.39002817847374732127208265908, 8.065713698206038211786760072955, 9.131470045751148029326261546587, 10.00973964275247819921783406647, 10.88460572419179190162204560888