| L(s) = 1 | − 7-s + 6·11-s − 13-s − 3·17-s − 2·19-s − 6·29-s + 4·31-s + 7·37-s − 43-s − 3·47-s − 6·49-s − 6·59-s + 8·61-s + 14·67-s − 3·71-s − 2·73-s − 6·77-s − 8·79-s − 12·83-s + 6·89-s + 91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.277·13-s − 0.727·17-s − 0.458·19-s − 1.11·29-s + 0.718·31-s + 1.15·37-s − 0.152·43-s − 0.437·47-s − 6/7·49-s − 0.781·59-s + 1.02·61-s + 1.71·67-s − 0.356·71-s − 0.234·73-s − 0.683·77-s − 0.900·79-s − 1.31·83-s + 0.635·89-s + 0.104·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.76574167210423, −14.44307305419407, −13.89724771484772, −13.20632058557853, −12.87294911773574, −12.30732528981569, −11.58489803878379, −11.42657571551557, −10.82293674676351, −9.987744625443318, −9.633589701948786, −9.143745636030326, −8.634883054978519, −8.054887167928469, −7.311793579573090, −6.749118595817576, −6.359085644224304, −5.869235247354877, −5.004492170261163, −4.347361831375907, −3.919857457242722, −3.273179066352187, −2.456276915175207, −1.754907782712230, −1.005447843208706, 0,
1.005447843208706, 1.754907782712230, 2.456276915175207, 3.273179066352187, 3.919857457242722, 4.347361831375907, 5.004492170261163, 5.869235247354877, 6.359085644224304, 6.749118595817576, 7.311793579573090, 8.054887167928469, 8.634883054978519, 9.143745636030326, 9.633589701948786, 9.987744625443318, 10.82293674676351, 11.42657571551557, 11.58489803878379, 12.30732528981569, 12.87294911773574, 13.20632058557853, 13.89724771484772, 14.44307305419407, 14.76574167210423
