L(s) = 1 | + 1.31·5-s + 4.77i·7-s − 6.27i·11-s − 8.24·17-s + 4.35i·19-s + 4i·23-s − 3.27·25-s + 6.27i·35-s + 7.40i·43-s + 10.2i·47-s − 15.8·49-s − 8.24i·55-s + 3.72·61-s − 16.8·73-s + 29.9·77-s + ⋯ |
L(s) = 1 | + 0.587·5-s + 1.80i·7-s − 1.89i·11-s − 1.99·17-s + 0.999i·19-s + 0.834i·23-s − 0.654·25-s + 1.06i·35-s + 1.12i·43-s + 1.49i·47-s − 2.26·49-s − 1.11i·55-s + 0.476·61-s − 1.96·73-s + 3.41·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8465700783\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8465700783\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 - 4.77iT - 7T^{2} \) |
| 11 | \( 1 + 6.27iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 8.24T + 17T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 7.40iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.052382850390418167827212415347, −8.546849775742501489124580453960, −7.88558836625414795432962437458, −6.52762724531045928923466439902, −5.86713822063660204283997131381, −5.68000963689963697677141984120, −4.50961838024423821353215567903, −3.30105627828781030324714710383, −2.54142078944707261768655395602, −1.62889865657664972490551608302,
0.24850038226883926577445575365, 1.72712436123262199917596823120, 2.48671475318642531732557436474, 4.06475995184773309152728200168, 4.37694104989853779844032962412, 5.20208038563415890930783015895, 6.68237833655866182565302784924, 6.83979927225961847070226307290, 7.51036126330667452596110196730, 8.566528748508040674822907405832