| L(s) = 1 | − i·5-s − 7-s + 3i·11-s − 5.29i·13-s + 5.29·17-s − 5.29i·19-s − 5.29·23-s + 4·25-s − 6i·29-s + 7·31-s + i·35-s − 5.29i·37-s − 5.29·41-s − 10.5i·43-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.377·7-s + 0.904i·11-s − 1.46i·13-s + 1.28·17-s − 1.21i·19-s − 1.10·23-s + 0.800·25-s − 1.11i·29-s + 1.25·31-s + 0.169i·35-s − 0.869i·37-s − 0.826·41-s − 1.61i·43-s − 0.857·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12853 - 0.779911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12853 - 0.779911i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 + 5.29iT - 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 5.29iT - 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 5.29iT - 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7iT - 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−10.03771931650392526662424647237, −9.269106172999229319230150311684, −8.206026077001193047690204757028, −7.58912552435414839162859848658, −6.53500292193328901558873171229, −5.52037682977658018657072183825, −4.74166544748677494774146752025, −3.53468971814356211826213266808, −2.41404401410024414695033866444, −0.71462573544980996180213854035,
1.47198860125743479685363774230, 3.00770398550170545518339886925, 3.80303822637974391952186199041, 5.05080893338026514397457181454, 6.20769466310527125688871923779, 6.67730787578043045712963933688, 7.895483296909452110606016240035, 8.544954386627162050391309658788, 9.688315227586014244368783178508, 10.16519987840079094404325674820
