L(s) = 1 | + (−1.5 − 0.866i)3-s + 1.73i·5-s − 1.73i·7-s + (1.5 + 2.59i)9-s − 13-s + (1.49 − 2.59i)15-s + 1.73i·17-s + 3.46i·19-s + (−1.49 + 2.59i)21-s + 6·23-s + 2.00·25-s − 5.19i·27-s + 6.92i·29-s + 3.46i·31-s + 2.99·35-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + 0.774i·5-s − 0.654i·7-s + (0.5 + 0.866i)9-s − 0.277·13-s + (0.387 − 0.670i)15-s + 0.420i·17-s + 0.794i·19-s + (−0.327 + 0.566i)21-s + 1.25·23-s + 0.400·25-s − 0.999i·27-s + 1.28i·29-s + 0.622i·31-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01958 + 0.273196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01958 + 0.273196i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 1.73iT - 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.69278005903977297106871428806, −10.23678175086013798097641279074, −8.949049855542413340453233797162, −7.67086701624874893266228282852, −7.10084586049746251980927493452, −6.31280552217635835200433696654, −5.30796703796158874061048958489, −4.19181892945315703687364056662, −2.82949436860675664543436214061, −1.22806379712622037220416208836,
0.77891088897599007283079227152, 2.69156371615424922345819539367, 4.28077129458403140817299104907, 5.00009315640912756947888693595, 5.81720253103164057555697317854, 6.79600227196598273710751089560, 7.947427797170295065900704670126, 9.246146081989699745371023490426, 9.368802879072057133690714691291, 10.66507103480687716051500418936