L(s) = 1 | + (3.35 + 1.93i)5-s + (2.5 + 0.866i)7-s + (3.35 − 1.93i)11-s + 3.46i·13-s + (2 − 3.46i)19-s + (−6.70 − 3.87i)23-s + (5.00 + 8.66i)25-s − 6.70·29-s + (−0.5 − 0.866i)31-s + (6.70 + 7.74i)35-s + (−2 + 3.46i)37-s − 7.74i·41-s − 6.92i·43-s + (−6.70 + 11.6i)47-s + (5.5 + 4.33i)49-s + ⋯ |
L(s) = 1 | + (1.50 + 0.866i)5-s + (0.944 + 0.327i)7-s + (1.01 − 0.583i)11-s + 0.960i·13-s + (0.458 − 0.794i)19-s + (−1.39 − 0.807i)23-s + (1.00 + 1.73i)25-s − 1.24·29-s + (−0.0898 − 0.155i)31-s + (1.13 + 1.30i)35-s + (−0.328 + 0.569i)37-s − 1.20i·41-s − 1.05i·43-s + (−0.978 + 1.69i)47-s + (0.785 + 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.385255340\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.385255340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-3.35 - 1.93i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.35 + 1.93i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.70 + 3.87i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.74iT - 41T^{2} \) |
| 43 | \( 1 + 6.92iT - 43T^{2} \) |
| 47 | \( 1 + (6.70 - 11.6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.35 + 5.80i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.35 + 5.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9 - 5.19i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 3.46i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.74iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.5 - 6.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + (-6.70 - 3.87i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−9.952906981415931709743554420855, −9.261941854307159834179319249370, −8.614445382978127797714007138123, −7.37619543517435020569116353335, −6.48814240683922879799795106919, −5.92297430354469545985470702092, −4.94654813181701953510236998979, −3.72506842785513394010179512699, −2.36129205700071365374079727117, −1.64274280156998604881553889558,
1.34475117445146891390206886479, 1.96455769706301333888722050995, 3.70426249213357988878858868000, 4.80952471233036566344778781263, 5.55454660153073689525482995804, 6.24160943624386512154633614810, 7.52248382687598867074062565644, 8.244479611085667202216951880944, 9.241266803621624269415961861260, 9.799266901010023586090539598515