L(s) = 1 | + (−0.511 − 0.137i)2-s − i·3-s + (−1.48 − 0.859i)4-s + (−2.03 + 0.545i)5-s + (−0.137 + 0.511i)6-s + (0.917 + 2.48i)7-s + (1.39 + 1.39i)8-s − 9-s + 1.11·10-s + (−2.03 − 2.03i)11-s + (−0.859 + 1.48i)12-s + (−2.80 + 2.27i)13-s + (−0.129 − 1.39i)14-s + (0.545 + 2.03i)15-s + (1.19 + 2.07i)16-s + (−1.64 + 2.85i)17-s + ⋯ |
L(s) = 1 | + (−0.361 − 0.0969i)2-s − 0.577i·3-s + (−0.744 − 0.429i)4-s + (−0.911 + 0.244i)5-s + (−0.0559 + 0.208i)6-s + (0.346 + 0.937i)7-s + (0.492 + 0.492i)8-s − 0.333·9-s + 0.353·10-s + (−0.614 − 0.614i)11-s + (−0.248 + 0.429i)12-s + (−0.776 + 0.629i)13-s + (−0.0345 − 0.373i)14-s + (0.140 + 0.526i)15-s + (0.299 + 0.518i)16-s + (−0.399 + 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.165366 + 0.235487i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.165366 + 0.235487i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (-0.917 - 2.48i)T \) |
| 13 | \( 1 + (2.80 - 2.27i)T \) |
good | 2 | \( 1 + (0.511 + 0.137i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (2.03 - 0.545i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (2.03 + 2.03i)T + 11iT^{2} \) |
| 17 | \( 1 + (1.64 - 2.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.78 - 4.78i)T + 19iT^{2} \) |
| 23 | \( 1 + (6.79 - 3.92i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.677 + 1.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.71 + 6.38i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 3.93i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (6.61 - 1.77i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (4.36 - 2.51i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.947 + 3.53i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.87 + 6.71i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.737 - 2.75i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 3.63iT - 61T^{2} \) |
| 67 | \( 1 + (1.33 - 1.33i)T - 67iT^{2} \) |
| 71 | \( 1 + (9.87 + 2.64i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-6.80 - 1.82i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (1.13 - 1.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.97 - 3.97i)T + 83iT^{2} \) |
| 89 | \( 1 + (-11.9 - 3.19i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.94 - 14.7i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.87469678664402078712874363504, −11.53865426625393739734083636721, −10.21017608140511300847224082852, −9.288608813424864184974422870012, −8.058570656912045795900613207021, −7.85823949947690488347676104792, −6.11418104369095144464867581662, −5.17910003470037117665575960667, −3.75430262859385268222145888721, −1.95715984636027796023124746700,
0.24842674189567236805730497158, 3.20225899484696934416755545910, 4.52904403846858530441430458521, 4.89630529691891196445488547162, 7.14863030361726471215347119714, 7.79617047822287870174712813633, 8.631121608153468018441574707734, 9.832092707267006338829878608752, 10.38037861564429822164946726096, 11.64595031727539354211231340377