| L(s) = 1 | − 7-s + 6·13-s − 2·19-s + 4·23-s − 5·25-s − 2·29-s − 2·31-s + 2·37-s − 8·41-s − 2·47-s + 49-s + 10·53-s − 4·59-s − 10·61-s − 4·67-s + 8·73-s + 8·79-s + 2·83-s + 6·89-s − 6·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.66·13-s − 0.458·19-s + 0.834·23-s − 25-s − 0.371·29-s − 0.359·31-s + 0.328·37-s − 1.24·41-s − 0.291·47-s + 1/7·49-s + 1.37·53-s − 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.936·73-s + 0.900·79-s + 0.219·83-s + 0.635·89-s − 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.62708560729385, −13.30485782332234, −13.05743815146249, −12.21988450767905, −11.98055264983642, −11.25214728675101, −10.90752385226145, −10.50685744937779, −9.916137158250649, −9.226336215709326, −9.051516140488792, −8.314680178742663, −8.028047919040380, −7.321937906673683, −6.716686683297193, −6.359073765631930, −5.749851363780097, −5.370301906902232, −4.564821647918125, −4.006600485644447, −3.490689575812487, −3.052052527693555, −2.169924933107851, −1.587909734717000, −0.8815176813799768, 0,
0.8815176813799768, 1.587909734717000, 2.169924933107851, 3.052052527693555, 3.490689575812487, 4.006600485644447, 4.564821647918125, 5.370301906902232, 5.749851363780097, 6.359073765631930, 6.716686683297193, 7.321937906673683, 8.028047919040380, 8.314680178742663, 9.051516140488792, 9.226336215709326, 9.916137158250649, 10.50685744937779, 10.90752385226145, 11.25214728675101, 11.98055264983642, 12.21988450767905, 13.05743815146249, 13.30485782332234, 13.62708560729385
