L(s) = 1 | + (2.74 − 2.74i)2-s + (−1.22 − 1.22i)3-s − 11.0i·4-s + (−0.683 + 4.95i)5-s − 6.71·6-s + (1.87 − 1.87i)7-s + (−19.3 − 19.3i)8-s + 2.99i·9-s + (11.7 + 15.4i)10-s + 10.9·11-s + (−13.5 + 13.5i)12-s + (8.10 + 8.10i)13-s − 10.2i·14-s + (6.90 − 5.22i)15-s − 61.7·16-s + (−5.51 + 5.51i)17-s + ⋯ |
L(s) = 1 | + (1.37 − 1.37i)2-s + (−0.408 − 0.408i)3-s − 2.76i·4-s + (−0.136 + 0.990i)5-s − 1.11·6-s + (0.267 − 0.267i)7-s + (−2.41 − 2.41i)8-s + 0.333i·9-s + (1.17 + 1.54i)10-s + 0.993·11-s + (−1.12 + 1.12i)12-s + (0.623 + 0.623i)13-s − 0.732i·14-s + (0.460 − 0.348i)15-s − 3.85·16-s + (−0.324 + 0.324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.981531 - 2.08460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.981531 - 2.08460i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 + (0.683 - 4.95i)T \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
good | 2 | \( 1 + (-2.74 + 2.74i)T - 4iT^{2} \) |
| 11 | \( 1 - 10.9T + 121T^{2} \) |
| 13 | \( 1 + (-8.10 - 8.10i)T + 169iT^{2} \) |
| 17 | \( 1 + (5.51 - 5.51i)T - 289iT^{2} \) |
| 19 | \( 1 + 12.1iT - 361T^{2} \) |
| 23 | \( 1 + (-24.3 - 24.3i)T + 529iT^{2} \) |
| 29 | \( 1 + 14.8iT - 841T^{2} \) |
| 31 | \( 1 + 8.07T + 961T^{2} \) |
| 37 | \( 1 + (34.6 - 34.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 32.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (13.0 + 13.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (54.1 - 54.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-6.76 - 6.76i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 44.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 84.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-0.661 + 0.661i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 103.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (55.1 + 55.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 68.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (71.4 + 71.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 41.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-25.2 + 25.2i)T - 9.40e3iT^{2} \) |
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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
−13.19515370456965870853260188566, −11.87173668160928241988635079206, −11.32990907239286613956031119771, −10.63663584107220964364593891742, −9.322432100750769685490491080330, −6.93088423719053082070126545603, −6.04370338523042149828408623791, −4.49636387665808648247144254951, −3.27820334716662310684689179509, −1.59476577704947311700677452057,
3.67723302349111736687802892516, 4.77930441905488120289941901749, 5.66246320305292975518146241346, 6.80046240012490756710408164004, 8.247409592425586461706624375316, 9.051710860406030582635476672285, 11.25375691662554222898711743838, 12.29321532284556433136105607570, 12.90641426323085343682108811234, 14.08746644354911169347699461642