| L(s) = 1 | − 4·7-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 4·23-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s + 4·59-s + 2·61-s − 8·67-s + 6·73-s + 16·77-s − 16·83-s + 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.977·67-s + 0.702·73-s + 1.82·77-s − 1.75·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.19701510067795, −15.86185513787513, −15.56934341078800, −14.66900859023834, −14.14467536099988, −13.48229851166950, −13.01115816400021, −12.55436871910433, −12.00660849333250, −11.34137133304873, −10.55921831715056, −9.926211443881808, −9.754328157190628, −9.082588908520566, −8.157886594649515, −7.637806017908158, −7.153052956995023, −6.198604493964262, −5.914749089365225, −5.102652942778825, −4.360976077208032, −3.487360202410436, −2.848508893878016, −2.350614579025015, −0.9361376447269563, 0,
0.9361376447269563, 2.350614579025015, 2.848508893878016, 3.487360202410436, 4.360976077208032, 5.102652942778825, 5.914749089365225, 6.198604493964262, 7.153052956995023, 7.637806017908158, 8.157886594649515, 9.082588908520566, 9.754328157190628, 9.926211443881808, 10.55921831715056, 11.34137133304873, 12.00660849333250, 12.55436871910433, 13.01115816400021, 13.48229851166950, 14.14467536099988, 14.66900859023834, 15.56934341078800, 15.86185513787513, 16.19701510067795
