L(s) = 1 | + 1.73i·5-s − 5.19i·11-s − 6.92i·13-s + 3.46i·17-s − 2·19-s + 6.92i·23-s + 2.00·25-s + 9·29-s − 31-s − 2·37-s − 3.46i·41-s − 3.46i·43-s − 9·53-s + 9·55-s − 3·59-s + ⋯ |
L(s) = 1 | + 0.774i·5-s − 1.56i·11-s − 1.92i·13-s + 0.840i·17-s − 0.458·19-s + 1.44i·23-s + 0.400·25-s + 1.67·29-s − 0.179·31-s − 0.328·37-s − 0.541i·41-s − 0.528i·43-s − 1.23·53-s + 1.21·55-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.306273504\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.306273504\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.902561194151981113991450976962, −7.02009639931977635766901108503, −6.24799844820075492541690941678, −5.73265661698145247940489403921, −5.10917350673191009348646510949, −3.89668402377204552347919205273, −3.16289797841770483105960018290, −2.83971158969057842666909100251, −1.43190693171559464666531507176, −0.33646107441497139943307893810,
1.14616200313891886986538043202, 2.02649648965153145572762321785, 2.77891514688239212122792227168, 4.15667341144945807492761991799, 4.69441245099543341064498590501, 4.86473286169586115822361043206, 6.26622755948430460701194563576, 6.74611835301507404003353578648, 7.30278816188348665662173856129, 8.254762292639315656057556791450