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
−9.879702338006211502008750907890, −8.882022699604063161175003990171, −8.501076689446542083857572270306, −7.27706027875578380231569869976, −6.52331597852173763444731355765, −5.79265631808348482552702571428, −4.62728601826417642329631424696, −3.70263642532890821053294044803, −2.62112419134848905602478919957, −1.24760914673966060729512368004,
0.914404346678317626253285303243, 2.24933474513764952603908938743, 3.76694970362761823829882908992, 4.25328063149764597039923097418, 5.64686861246427816297408382419, 6.24315695442239924398328202053, 7.29447520560921211994125246126, 8.119663144162866958006445141190, 8.875362810926045365395798507057, 9.776044519940063914114074583613