L(s) = 1 | − 5-s + 0.286i·11-s + 6.53i·13-s − 1.74·17-s + 2.17i·19-s − 2.66i·23-s + 25-s + 4.02i·29-s + 8.98i·31-s − 3.68·37-s − 9.19·41-s − 6.70·43-s + 11.4·47-s − 5.29i·53-s − 0.286i·55-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.0863i·11-s + 1.81i·13-s − 0.422·17-s + 0.499i·19-s − 0.555i·23-s + 0.200·25-s + 0.746i·29-s + 1.61i·31-s − 0.606·37-s − 1.43·41-s − 1.02·43-s + 1.66·47-s − 0.727i·53-s − 0.0385i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2032540867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2032540867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.286iT - 11T^{2} \) |
| 13 | \( 1 - 6.53iT - 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 - 2.17iT - 19T^{2} \) |
| 23 | \( 1 + 2.66iT - 23T^{2} \) |
| 29 | \( 1 - 4.02iT - 29T^{2} \) |
| 31 | \( 1 - 8.98iT - 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + 9.19T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 5.29iT - 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 0.709T + 67T^{2} \) |
| 71 | \( 1 + 7.64iT - 71T^{2} \) |
| 73 | \( 1 + 4.28iT - 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 - 4.03T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 + 1.29iT - 97T^{2} \) |
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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
−8.219832415796963501108872561482, −7.34259837802287889686579236425, −6.71319965932809969819693525842, −6.39989520700984410625216541980, −5.15279031187546315843864286236, −4.73810075853085891554059916904, −3.87045683083847441842578698270, −3.28734550413966289943079576782, −2.12220333950259182050293188751, −1.45585146461531174200398765782,
0.05198036614835496817113155921, 0.978150336955341883212302911253, 2.27298836006486823725495668523, 3.02560083681081539247416945911, 3.76922042936352670722296908407, 4.52434106813245145989401525633, 5.40620839036223090689292111335, 5.84198504498144598970058980093, 6.75930270842137803298053880931, 7.48834148349950863978057856386