| L(s) = 1 | + 2.13e3·5-s + 6.32e3·7-s − 7.27e4·11-s + 1.45e5·13-s − 1.98e5·17-s − 2.35e5·19-s − 1.02e6·23-s + 2.61e6·25-s − 5.88e6·29-s − 9.92e6·31-s + 1.35e7·35-s − 1.75e7·37-s − 4.41e6·41-s − 3.21e7·43-s − 4.32e7·47-s − 3.48e5·49-s − 7.26e6·53-s − 1.55e8·55-s − 1.86e7·59-s + 1.48e8·61-s + 3.10e8·65-s + 2.02e8·67-s + 2.35e8·71-s − 1.13e8·73-s − 4.60e8·77-s + 2.89e8·79-s − 3.59e8·83-s + ⋯ |
| L(s) = 1 | + 1.52·5-s + 0.995·7-s − 1.49·11-s + 1.41·13-s − 0.575·17-s − 0.414·19-s − 0.760·23-s + 1.34·25-s − 1.54·29-s − 1.93·31-s + 1.52·35-s − 1.54·37-s − 0.244·41-s − 1.43·43-s − 1.29·47-s − 0.00864·49-s − 0.126·53-s − 2.29·55-s − 0.200·59-s + 1.37·61-s + 2.16·65-s + 1.22·67-s + 1.10·71-s − 0.466·73-s − 1.49·77-s + 0.835·79-s − 0.832·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 2.13e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.32e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.27e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.45e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.98e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.02e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.88e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.75e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.41e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.32e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.26e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.86e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.48e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.02e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.35e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.13e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.30e7T + 7.60e17T^{2} \) |
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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
−9.844869913529150356677444690234, −8.773221339646603808521637053133, −8.021397108929883866197449524330, −6.71159213044184920291733897219, −5.60354134934156515126566768057, −5.13300591683856970138692834574, −3.58530495862992481381253870542, −2.03166374810893924898174771410, −1.70918195596791924817448053798, 0,
1.70918195596791924817448053798, 2.03166374810893924898174771410, 3.58530495862992481381253870542, 5.13300591683856970138692834574, 5.60354134934156515126566768057, 6.71159213044184920291733897219, 8.021397108929883866197449524330, 8.773221339646603808521637053133, 9.844869913529150356677444690234
