| L(s) = 1 | − i·2-s − 4-s + 5-s − 2.68i·7-s + i·8-s − i·10-s + 1.73·11-s − 2.55·13-s − 2.68·14-s + 16-s + 1.27·17-s − 1.40i·19-s − 20-s − 1.73i·22-s + (−4.59 − 1.35i)23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447·5-s − 1.01i·7-s + 0.353i·8-s − 0.316i·10-s + 0.521·11-s − 0.708·13-s − 0.718·14-s + 0.250·16-s + 0.308·17-s − 0.322i·19-s − 0.223·20-s − 0.369i·22-s + (−0.958 − 0.283i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294465837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294465837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (4.59 + 1.35i)T \) |
| good | 7 | \( 1 + 2.68iT - 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 + 1.40iT - 19T^{2} \) |
| 29 | \( 1 + 9.62iT - 29T^{2} \) |
| 31 | \( 1 + 1.47T + 31T^{2} \) |
| 37 | \( 1 - 4.20iT - 37T^{2} \) |
| 41 | \( 1 + 3.40iT - 41T^{2} \) |
| 43 | \( 1 + 2.35iT - 43T^{2} \) |
| 47 | \( 1 + 3.21iT - 47T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 + 7.74iT - 59T^{2} \) |
| 61 | \( 1 - 6.78iT - 61T^{2} \) |
| 67 | \( 1 - 7.71iT - 67T^{2} \) |
| 71 | \( 1 + 4.44iT - 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 + 10.0iT - 79T^{2} \) |
| 83 | \( 1 + 8.67T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 - 7.03iT - 97T^{2} \) |
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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
−8.947455609607528038135285832009, −8.038617373348454444023898560058, −7.27648157257700152034837893807, −6.40700132137174609238873836026, −5.48439437851036014285344486207, −4.44616670176825475962816995482, −3.87260971201880854733515701081, −2.71566831222279973818818641889, −1.71695228659978005802234792006, −0.45643038304145743228205748372,
1.53922462560733713374670401107, 2.69423254933840803225563159868, 3.80596754837312023872675653378, 4.92691814994740456156901265651, 5.58609203718777254784431579386, 6.24534452960828756524433084612, 7.08674371475629857969270988032, 7.897269239603651747758440526131, 8.730603698478087148794163567388, 9.330074453749486527026452046546
