L(s) = 1 | + (0.719 + 1.21i)2-s + (−1.66 − 1.66i)3-s + (−0.965 + 1.75i)4-s + (0.831 − 3.23i)6-s − 1.87i·7-s + (−2.82 + 0.0835i)8-s + 2.56i·9-s + (−3.29 + 3.29i)11-s + (4.53 − 1.31i)12-s + (−1.90 − 1.90i)13-s + (2.28 − 1.34i)14-s + (−2.13 − 3.38i)16-s − 2.57·17-s + (−3.12 + 1.84i)18-s + (−5.76 − 5.76i)19-s + ⋯ |
L(s) = 1 | + (0.508 + 0.861i)2-s + (−0.962 − 0.962i)3-s + (−0.482 + 0.875i)4-s + (0.339 − 1.31i)6-s − 0.708i·7-s + (−0.999 + 0.0295i)8-s + 0.854i·9-s + (−0.994 + 0.994i)11-s + (1.30 − 0.378i)12-s + (−0.527 − 0.527i)13-s + (0.609 − 0.360i)14-s + (−0.533 − 0.845i)16-s − 0.623·17-s + (−0.735 + 0.434i)18-s + (−1.32 − 1.32i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.149478 - 0.288631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149478 - 0.288631i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.719 - 1.21i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.66 + 1.66i)T + 3iT^{2} \) |
| 7 | \( 1 + 1.87iT - 7T^{2} \) |
| 11 | \( 1 + (3.29 - 3.29i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.90 + 1.90i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 + (5.76 + 5.76i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.58iT - 23T^{2} \) |
| 29 | \( 1 + (-6.45 - 6.45i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.799T + 31T^{2} \) |
| 37 | \( 1 + (2.69 - 2.69i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.946iT - 41T^{2} \) |
| 43 | \( 1 + (0.829 - 0.829i)T - 43iT^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + (6.97 - 6.97i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.84 + 6.84i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.87 + 6.87i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.73 - 3.73i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.34iT - 71T^{2} \) |
| 73 | \( 1 + 0.886iT - 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 + (-0.989 - 0.989i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−11.04846290520601843510343870271, −10.28416672572298235874538893207, −8.772538600121928956291574123347, −7.76302945311117717889705693402, −6.86064091030075635864797642603, −6.54472526960342308365685573349, −5.11047498309009773161986457257, −4.52255054817469611639993411011, −2.57090940656335871593130085736, −0.18905210829248421287678108721,
2.27030186678085329566053132776, 3.71579863359804070657718299467, 4.74414880102722770565877603880, 5.60358420662000016121263790412, 6.22733604098018534648038737577, 8.182236102276638195874121425999, 9.239820184433477162529677905401, 10.15790318937671452828148467770, 10.73122332331200310315777528051, 11.56005931544231653358255311170