L(s) = 1 | + (−0.690 − 2.12i)2-s + (1 + 0.726i)3-s + (−2.42 + 1.76i)4-s + (−0.618 + 1.90i)5-s + (0.854 − 2.62i)6-s + (0.809 − 0.587i)7-s + (1.80 + 1.31i)8-s + (−0.454 − 1.40i)9-s + 4.47·10-s − 3.70·12-s + (−1 − 3.07i)13-s + (−1.80 − 1.31i)14-s + (−2 + 1.45i)15-s + (−0.309 + 0.951i)16-s + (1 − 3.07i)17-s + (−2.66 + 1.93i)18-s + ⋯ |
L(s) = 1 | + (−0.488 − 1.50i)2-s + (0.577 + 0.419i)3-s + (−1.21 + 0.881i)4-s + (−0.276 + 0.850i)5-s + (0.348 − 1.07i)6-s + (0.305 − 0.222i)7-s + (0.639 + 0.464i)8-s + (−0.151 − 0.466i)9-s + 1.41·10-s − 1.07·12-s + (−0.277 − 0.853i)13-s + (−0.483 − 0.351i)14-s + (−0.516 + 0.375i)15-s + (−0.0772 + 0.237i)16-s + (0.242 − 0.746i)17-s + (−0.627 + 0.456i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.676565 - 1.08205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.676565 - 1.08205i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.690 + 2.12i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1 - 0.726i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (0.618 - 1.90i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (1 + 3.07i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1 + 3.07i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.23 - 3.80i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + (-6.85 + 4.97i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.854 + 2.62i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.85 + 4.97i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.09 + 6.60i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (2.23 + 1.62i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.145 + 0.449i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1 + 0.726i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.23 + 6.88i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + (3.23 - 9.95i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.618 - 0.449i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.76 - 8.50i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.52 + 10.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (-5.38 - 16.5i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.967821626384007363028151037940, −9.502809583374590854212084170944, −8.449397600813701704478660058939, −7.73519589734094867385253721065, −6.61236960017253930867164727758, −5.18157308281380591939021325858, −3.81542633798593752223528709530, −3.23651297372140709096408757499, −2.46910542114465016079972906756, −0.793923514844864063011695639305,
1.31189758006944710991201326104, 2.95768035574715319646249054185, 4.83624167887965794908445628535, 5.06340224851457316106961870832, 6.46233912785360185458527774648, 7.14683455222129430956120164306, 8.049742524054736301640535806846, 8.506060145418433489006908949099, 9.084696981601805323306179063069, 10.01545860658817049382011052428