L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s + 2.75·7-s − i·8-s + 9-s − 10-s + 4.75·11-s − 12-s − 6.75i·13-s + 2.75i·14-s + i·15-s + 16-s − 7.06i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 1.03·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.43·11-s − 0.288·12-s − 1.87i·13-s + 0.735i·14-s + 0.258i·15-s + 0.250·16-s − 1.71i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.277291278\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.277291278\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (4.90 - 3.59i)T \) |
good | 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 + 6.75iT - 13T^{2} \) |
| 17 | \( 1 + 7.06iT - 17T^{2} \) |
| 19 | \( 1 - 6.75iT - 19T^{2} \) |
| 23 | \( 1 + 5.06iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 0.315iT - 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.81iT - 43T^{2} \) |
| 47 | \( 1 - 3.50T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 - 11.8iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 6.43T + 73T^{2} \) |
| 79 | \( 1 - 8.31iT - 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 + 6.25iT - 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−9.857507936389302759441045408944, −8.888890386349630418019849483327, −8.126365975647094721634026138577, −7.60268433864957250425172581591, −6.69234505785433429918150981027, −5.72927241026616397126176919862, −4.80265329687410360886994779318, −3.79409821659877342865488685514, −2.74710474826647008075535668407, −1.16310788075165819611705531178,
1.48933829900666965542041427950, 1.97403564587966968325803893320, 3.73213394971237129634747897045, 4.21670720821100285176876464842, 5.15865534989914161451649026893, 6.51134889635301615228005387156, 7.34639350852534140463883131247, 8.635275026505422492192058799438, 8.845323486298143412218666662312, 9.551914491491157003672155178748