L(s) = 1 | − 3-s − 4·5-s − 2·9-s − 3·11-s − 5·13-s + 4·15-s + 17-s − 6·19-s + 4·23-s + 11·25-s + 5·27-s − 8·29-s + 3·33-s + 8·37-s + 5·39-s + 8·41-s + 10·43-s + 8·45-s − 2·47-s − 51-s − 3·53-s + 12·55-s + 6·57-s − 2·59-s − 8·61-s + 20·65-s + 14·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.904·11-s − 1.38·13-s + 1.03·15-s + 0.242·17-s − 1.37·19-s + 0.834·23-s + 11/5·25-s + 0.962·27-s − 1.48·29-s + 0.522·33-s + 1.31·37-s + 0.800·39-s + 1.24·41-s + 1.52·43-s + 1.19·45-s − 0.291·47-s − 0.140·51-s − 0.412·53-s + 1.61·55-s + 0.794·57-s − 0.260·59-s − 1.02·61-s + 2.48·65-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.98315409147031, −14.56078423670992, −14.23101227360606, −13.11263989619804, −12.67225757714595, −12.51375290740198, −11.86039971054657, −11.23488137318044, −11.03210899851035, −10.66617991779924, −9.812890306345039, −9.215445116417312, −8.610176929211600, −8.009622777933602, −7.639030965605177, −7.225805870205977, −6.578635597773967, −5.768124240941405, −5.342411933902495, −4.489025600816516, −4.393206195874410, −3.511267501004365, −2.726251095400751, −2.427864208599047, −1.017518343482637, 0, 0,
1.017518343482637, 2.427864208599047, 2.726251095400751, 3.511267501004365, 4.393206195874410, 4.489025600816516, 5.342411933902495, 5.768124240941405, 6.578635597773967, 7.225805870205977, 7.639030965605177, 8.009622777933602, 8.610176929211600, 9.215445116417312, 9.812890306345039, 10.66617991779924, 11.03210899851035, 11.23488137318044, 11.86039971054657, 12.51375290740198, 12.67225757714595, 13.11263989619804, 14.23101227360606, 14.56078423670992, 14.98315409147031