L(s) = 1 | + (1.41 − 1.73i)5-s + 2.44·7-s − 1.41i·11-s − 2.44·13-s − 3.46·17-s − 6·19-s − 6.92i·23-s + (−0.999 − 4.89i)25-s − 5.65·29-s + 2i·31-s + (3.46 − 4.24i)35-s − 7.34·37-s − 4.24i·41-s − 4.89i·43-s + 3.46i·47-s + ⋯ |
L(s) = 1 | + (0.632 − 0.774i)5-s + 0.925·7-s − 0.426i·11-s − 0.679·13-s − 0.840·17-s − 1.37·19-s − 1.44i·23-s + (−0.199 − 0.979i)25-s − 1.05·29-s + 0.359i·31-s + (0.585 − 0.717i)35-s − 1.20·37-s − 0.662i·41-s − 0.747i·43-s + 0.505i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.251887115\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.251887115\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 + 1.73i)T \) |
good | 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 + 1.41iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 + 14.6iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−8.682906088062632810622742240110, −7.915664564182624168692975431158, −6.88341371243266767806184959231, −6.18529426896712875194327532148, −5.21425251925022154243467849567, −4.72564087234748530806038214648, −3.88868296281029900507243216151, −2.37392418908361656493572057160, −1.81553402634386337737615904284, −0.35256959671969782372835165590,
1.75043768036228261627317813180, 2.23142797406023346227331038619, 3.43689164368790767464992377549, 4.45605210295460851253519165137, 5.19772512383969025664535423671, 6.04184163440589944399161652925, 6.85950562470236043433342289253, 7.48179926283403750092479206257, 8.241346541526591943861310250800, 9.181883757371710668080770763500