L(s) = 1 | + (−3.76 + 1.36i)2-s + (12.2 − 10.2i)4-s + 28.6·5-s + 25.6i·7-s + (−32.3 + 55.2i)8-s + (−107. + 38.9i)10-s − 108. i·11-s − 88.4·13-s + (−34.9 − 96.4i)14-s + (46.2 − 251. i)16-s − 504.·17-s + 191. i·19-s + (351. − 292. i)20-s + (147. + 408. i)22-s − 960. i·23-s + ⋯ |
L(s) = 1 | + (−0.940 + 0.340i)2-s + (0.768 − 0.640i)4-s + 1.14·5-s + 0.523i·7-s + (−0.504 + 0.863i)8-s + (−1.07 + 0.389i)10-s − 0.896i·11-s − 0.523·13-s + (−0.178 − 0.492i)14-s + (0.180 − 0.983i)16-s − 1.74·17-s + 0.530i·19-s + (0.879 − 0.732i)20-s + (0.305 + 0.843i)22-s − 1.81i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2899466783\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2899466783\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.76 - 1.36i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 28.6T + 625T^{2} \) |
| 7 | \( 1 - 25.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 108. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 88.4T + 2.85e4T^{2} \) |
| 17 | \( 1 + 504.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 191. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 960. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 793.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 329. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 209.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.05e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.33e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.12e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.13e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.66e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.59e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.07e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.43e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.95e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.75e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.02e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 559.T + 6.27e7T^{2} \) |
| 97 | \( 1 + 2.20e3T + 8.85e7T^{2} \) |
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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
−10.51965380828554050892183543280, −9.457429019839200727896652122068, −8.902120555981677466036401286931, −7.957914845077259404110006504666, −6.51317507877963565053965786491, −6.08145994631912121749472328762, −4.87927967244931766141963030843, −2.70661086447269515919031758406, −1.78466954416757516367411967987, −0.10632023959635357056901210690,
1.61724515692606520151766689334, 2.43910351354366019696015728595, 4.08610675373486878354869799965, 5.56607634389628615933952859169, 6.86706329775650952691402098706, 7.41646956508465105210314679585, 8.878705854963584744102851609752, 9.524482868712665553023451971819, 10.22601655507968858364516243262, 11.11131577415668583691275541918