| L(s) = 1 | + 2.96i·5-s + i·7-s − 4.76·11-s − 1.26·13-s − 0.378i·17-s + 5i·19-s + 1.93·23-s − 3.80·25-s + 4.24i·29-s + 10.4i·31-s − 2.96·35-s + 2.46·37-s − 10.6i·41-s + 4.73i·43-s + 3.20·47-s + ⋯ |
| L(s) = 1 | + 1.32i·5-s + 0.377i·7-s − 1.43·11-s − 0.351·13-s − 0.0919i·17-s + 1.14i·19-s + 0.402·23-s − 0.760·25-s + 0.787i·29-s + 1.87i·31-s − 0.501·35-s + 0.405·37-s − 1.67i·41-s + 0.721i·43-s + 0.467·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4957086884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4957086884\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.96iT - 5T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 0.378iT - 17T^{2} \) |
| 19 | \( 1 - 5iT - 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 4.73iT - 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 + 8.76iT - 53T^{2} \) |
| 59 | \( 1 + 3.48T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 7.66iT - 67T^{2} \) |
| 71 | \( 1 - 0.240T + 71T^{2} \) |
| 73 | \( 1 + 0.196T + 73T^{2} \) |
| 79 | \( 1 - 1.80iT - 79T^{2} \) |
| 83 | \( 1 + 7.82T + 83T^{2} \) |
| 89 | \( 1 - 1.93iT - 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.364822958148990560358458538574, −7.74923511688820405377326287370, −7.05827270241831302974118642538, −6.56275735954465721580596350881, −5.52621303673630351654347766292, −5.19928683252745332354163777524, −4.01100066078243090766836188822, −3.04040113124639910722115681060, −2.70822838627461790142793633777, −1.64504207121137892434809411434,
0.13759393097661429038370973535, 0.992094567855113106685363102484, 2.24672272708011926796611399404, 2.97921419108854785323029544242, 4.30946979984740122694016563678, 4.63390326511198176861026991751, 5.39945779385487968811712299465, 6.03953620233166411432645803549, 7.07500472646684479172198611730, 7.85718450759317129124796770980
