L(s) = 1 | − 0.585i·5-s − 0.828i·7-s + 2.82·11-s + 2.82·13-s − 2.58i·17-s + 5.65i·19-s − 6.82·23-s + 4.65·25-s − 3.41i·29-s − 8.82i·31-s − 0.485·35-s + 7.65·37-s − 5.41i·41-s + 1.65i·43-s + 4.48·47-s + ⋯ |
L(s) = 1 | − 0.261i·5-s − 0.313i·7-s + 0.852·11-s + 0.784·13-s − 0.627i·17-s + 1.29i·19-s − 1.42·23-s + 0.931·25-s − 0.634i·29-s − 1.58i·31-s − 0.0820·35-s + 1.25·37-s − 0.845i·41-s + 0.252i·43-s + 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.728452146\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.728452146\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.585iT - 5T^{2} \) |
| 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.58iT - 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41iT - 29T^{2} \) |
| 31 | \( 1 + 8.82iT - 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 5.41iT - 41T^{2} \) |
| 43 | \( 1 - 1.65iT - 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 + 9.07iT - 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 10.4iT - 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−9.776044519940063914114074583613, −8.875362810926045365395798507057, −8.119663144162866958006445141190, −7.29447520560921211994125246126, −6.24315695442239924398328202053, −5.64686861246427816297408382419, −4.25328063149764597039923097418, −3.76694970362761823829882908992, −2.24933474513764952603908938743, −0.914404346678317626253285303243,
1.24760914673966060729512368004, 2.62112419134848905602478919957, 3.70263642532890821053294044803, 4.62728601826417642329631424696, 5.79265631808348482552702571428, 6.52331597852173763444731355765, 7.27706027875578380231569869976, 8.501076689446542083857572270306, 8.882022699604063161175003990171, 9.879702338006211502008750907890