L(s) = 1 | − 2.23·5-s + 4.47i·7-s − 2i·11-s + 4.47·13-s − 6i·19-s − 4.47i·23-s + 5.00·25-s − 10.0i·35-s + 4.47·37-s − 2·41-s + 13.4i·47-s − 13.0·49-s + 13.4·53-s + 4.47i·55-s + 14i·59-s + ⋯ |
L(s) = 1 | − 0.999·5-s + 1.69i·7-s − 0.603i·11-s + 1.24·13-s − 1.37i·19-s − 0.932i·23-s + 1.00·25-s − 1.69i·35-s + 0.735·37-s − 0.312·41-s + 1.95i·47-s − 1.85·49-s + 1.84·53-s + 0.603i·55-s + 1.82i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.481822137\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.481822137\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.23T \) |
good | 7 | \( 1 - 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.4iT - 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.773842422973516819954950297470, −8.363050337840027462126353052629, −7.45448961076620478387530869579, −6.44235506093297765595765324656, −5.90348127001235365034730919965, −4.98295677027111257577233644079, −4.14216891599355548815810148270, −3.09896336559619831676159268374, −2.46947881859814603717377037249, −0.883529440734440994670781926090,
0.67225649542520677441709668421, 1.70029375174573748636897083277, 3.49452582016570019424655434968, 3.77211831745869491915430860539, 4.51545136128790332904853534918, 5.58195784619286731690156618089, 6.66525620344050013213462002644, 7.20594128463520858240064071519, 7.927472281453022060639061994230, 8.394179263934849338485432317251