L(s) = 1 | + (1 − 1.41i)3-s + 1.41i·5-s + 4.24i·7-s + (−1.00 − 2.82i)9-s + 4·11-s − 2·13-s + (2.00 + 1.41i)15-s + 2.82i·17-s + (6 + 4.24i)21-s − 8·23-s + 2.99·25-s + (−5.00 − 1.41i)27-s + 7.07i·29-s − 1.41i·31-s + (4 − 5.65i)33-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s + 0.632i·5-s + 1.60i·7-s + (−0.333 − 0.942i)9-s + 1.20·11-s − 0.554·13-s + (0.516 + 0.365i)15-s + 0.685i·17-s + (1.30 + 0.925i)21-s − 1.66·23-s + 0.599·25-s + (−0.962 − 0.272i)27-s + 1.31i·29-s − 0.254i·31-s + (0.696 − 0.984i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.920381071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.920381071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 7.07iT - 29T^{2} \) |
| 31 | \( 1 + 1.41iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 5.65iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 9.89iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−9.335300707035135087291355864770, −8.783692027703141344389059633416, −8.037139763544219135152820247063, −7.17547761771272499724616523160, −6.17618759094197419331948350305, −5.98013654148060899319220091677, −4.44886709631316781002904232176, −3.25221795955999868860712157259, −2.48921298305731518201139172274, −1.55240907410029757448220024704,
0.71694642907932362836345352003, 2.19696243349823355194151207016, 3.64290288522659867770835178310, 4.18781973327505542717922223730, 4.81633408112939146916975300672, 6.02565924074058920417525803013, 7.13370363345745822617583670742, 7.77752118585259895470349953021, 8.628050543928402731924849008099, 9.485755890521220712659929235321