| 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
−11.11131577415668583691275541918, −10.22601655507968858364516243262, −9.524482868712665553023451971819, −8.878705854963584744102851609752, −7.41646956508465105210314679585, −6.86706329775650952691402098706, −5.56607634389628615933952859169, −4.08610675373486878354869799965, −2.43910351354366019696015728595, −1.61724515692606520151766689334,
0.10632023959635357056901210690, 1.78466954416757516367411967987, 2.70661086447269515919031758406, 4.87927967244931766141963030843, 6.08145994631912121749472328762, 6.51317507877963565053965786491, 7.957914845077259404110006504666, 8.902120555981677466036401286931, 9.457429019839200727896652122068, 10.51965380828554050892183543280
