L(s) = 1 | + (−0.464 + 0.124i)2-s + (−0.258 + 0.965i)3-s + (−1.53 + 0.884i)4-s − 0.480i·6-s + (−2.38 − 1.13i)7-s + (1.28 − 1.28i)8-s + (−0.866 − 0.499i)9-s + (1.38 + 2.39i)11-s + (−0.457 − 1.70i)12-s + (−0.707 − 0.707i)13-s + (1.25 + 0.231i)14-s + (1.33 − 2.30i)16-s + (−4.28 − 1.14i)17-s + (0.464 + 0.124i)18-s + (0.280 − 0.486i)19-s + ⋯ |
L(s) = 1 | + (−0.328 + 0.0880i)2-s + (−0.149 + 0.557i)3-s + (−0.765 + 0.442i)4-s − 0.196i·6-s + (−0.902 − 0.430i)7-s + (0.453 − 0.453i)8-s + (−0.288 − 0.166i)9-s + (0.417 + 0.722i)11-s + (−0.132 − 0.493i)12-s + (−0.196 − 0.196i)13-s + (0.334 + 0.0618i)14-s + (0.333 − 0.577i)16-s + (−1.03 − 0.278i)17-s + (0.109 + 0.0293i)18-s + (0.0643 − 0.111i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.296629 - 0.256883i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.296629 - 0.256883i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.38 + 1.13i)T \) |
good | 2 | \( 1 + (0.464 - 0.124i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 2.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.28 + 1.14i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.280 + 0.486i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.08 + 7.77i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 5.53iT - 29T^{2} \) |
| 31 | \( 1 + (-6.97 + 4.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.09 + 1.09i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.21iT - 41T^{2} \) |
| 43 | \( 1 + (-1.08 + 1.08i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.26 + 8.46i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.727 + 0.194i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.09 + 8.81i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.58 - 9.65i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + (2.03 - 7.60i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.05 + 4.65i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.20 + 6.20i)T + 83iT^{2} \) |
| 89 | \( 1 + (7.67 - 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.41 + 6.41i)T - 97iT^{2} \) |
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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
−10.24012062406340126891612818859, −9.833314099100014149002379586942, −9.028139898352806310064464393250, −8.134803942761129535286248568563, −7.03934994751527769802679535172, −6.16645060560348246475420779055, −4.56593799104403385811025961414, −4.17690078962958234649121257601, −2.76813831258500970483589969325, −0.27970190908609294829748080097,
1.44191811439055449998496605970, 3.08855391963945889657424907992, 4.44185841100889404992158212083, 5.71069036566267739900814238419, 6.33929706204644327768913740790, 7.50883738884128068687738232923, 8.673891546551313349384259544244, 9.155475760312240010845077903778, 10.07169817037521619025393539095, 11.00999848617233332179646619241