L(s) = 1 | + 1.73i·5-s + 1.73i·7-s − 3·11-s + 5·13-s + 6.92i·17-s − 3.46i·19-s + 9·23-s + 2.00·25-s − 1.73i·29-s − 5.19i·31-s − 2.99·35-s − 2·37-s + 5.19i·41-s + 5.19i·43-s + 3·47-s + ⋯ |
L(s) = 1 | + 0.774i·5-s + 0.654i·7-s − 0.904·11-s + 1.38·13-s + 1.68i·17-s − 0.794i·19-s + 1.87·23-s + 0.400·25-s − 0.321i·29-s − 0.933i·31-s − 0.507·35-s − 0.328·37-s + 0.811i·41-s + 0.792i·43-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.979200472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.979200472\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 9T + 23T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 8.66iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8.66iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−8.465228881280003051578712643536, −7.72323171442808617548991375098, −6.84645472681785116933849835324, −6.24778771839563916872662210302, −5.62516165285240581875011394205, −4.77871361540091166158362167198, −3.77751958887345520929614762693, −3.00719446299397611625270487578, −2.31659759631235003652655109615, −1.10181115834356533336870162408,
0.63295805193117981292190926546, 1.36142583811054545957352483151, 2.73488000083933898891814037984, 3.49281637379518849996451206796, 4.40117098225425295021103046427, 5.22441423566521426453473919341, 5.56302419554584969429267005843, 6.89850090292395935613614161426, 7.14596048291568799543740861323, 8.139653863252550983731007187116