| L(s) = 1 | − 4·5-s + 7-s + 11-s + 2·13-s + 2·19-s − 4·23-s + 11·25-s + 6·29-s − 10·31-s − 4·35-s − 2·37-s − 8·41-s − 2·47-s + 49-s + 6·53-s − 4·55-s + 8·59-s + 10·61-s − 8·65-s + 12·67-s − 8·71-s + 8·73-s + 77-s − 8·79-s − 10·83-s + 6·89-s + 2·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s + 0.458·19-s − 0.834·23-s + 11/5·25-s + 1.11·29-s − 1.79·31-s − 0.676·35-s − 0.328·37-s − 1.24·41-s − 0.291·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 1.04·59-s + 1.28·61-s − 0.992·65-s + 1.46·67-s − 0.949·71-s + 0.936·73-s + 0.113·77-s − 0.900·79-s − 1.09·83-s + 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.91566435459726, −14.48412319726370, −14.04848266985359, −13.31570794846258, −12.75795334172854, −12.19645777829551, −11.72796081892881, −11.45226154313807, −10.90049402335377, −10.33754238542272, −9.726915712390483, −8.880493344651070, −8.484704373760580, −8.098181460728927, −7.509443077500905, −6.947808762815340, −6.554934375971141, −5.521849172689019, −5.128950483540315, −4.295640986604481, −3.830643041667274, −3.481777165105746, −2.629442691741075, −1.667152692118703, −0.8538253373211856, 0,
0.8538253373211856, 1.667152692118703, 2.629442691741075, 3.481777165105746, 3.830643041667274, 4.295640986604481, 5.128950483540315, 5.521849172689019, 6.554934375971141, 6.947808762815340, 7.509443077500905, 8.098181460728927, 8.484704373760580, 8.880493344651070, 9.726915712390483, 10.33754238542272, 10.90049402335377, 11.45226154313807, 11.72796081892881, 12.19645777829551, 12.75795334172854, 13.31570794846258, 14.04848266985359, 14.48412319726370, 14.91566435459726
