L(s) = 1 | − i·3-s − i·5-s + 2·7-s + 2·9-s − i·11-s + 4i·13-s − 15-s − 2·17-s − 2i·21-s + 23-s + 4·25-s − 5i·27-s + 7·31-s − 33-s − 2i·35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 0.755·7-s + 0.666·9-s − 0.301i·11-s + 1.10i·13-s − 0.258·15-s − 0.485·17-s − 0.436i·21-s + 0.208·23-s + 0.800·25-s − 0.962i·27-s + 1.25·31-s − 0.174·33-s − 0.338i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.243666264\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.243666264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 5iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−8.687592205667708506770154565991, −7.916878178629849818679257198326, −7.20266238781882351223882659480, −6.53660396588792702119274524157, −5.69133036420126825100930033639, −4.45298646865807577967014499988, −4.38891509824061165744603971286, −2.79179970004899877947433686724, −1.77049251638321865536227064279, −0.951595236594138342146761919373,
1.03770031424667030589951552802, 2.32943227969259500862063844775, 3.26774570394034284914638144952, 4.28839229713789975173015221894, 4.86319187009344073465388727768, 5.69417433013758956753560108064, 6.72927831674085923018263854945, 7.37157487360820519231106280778, 8.162704554022076151146745808998, 8.858569439647824559752818618109