| L(s) = 1 | − 5-s − 2·7-s + 2·13-s + 2·19-s + 25-s + 8·31-s + 2·35-s − 2·37-s + 2·43-s − 3·49-s + 6·53-s − 12·59-s + 2·61-s − 2·65-s + 4·67-s − 2·73-s + 10·79-s + 12·83-s + 6·89-s − 4·91-s − 2·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.554·13-s + 0.458·19-s + 1/5·25-s + 1.43·31-s + 0.338·35-s − 0.328·37-s + 0.304·43-s − 3/7·49-s + 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.234·73-s + 1.12·79-s + 1.31·83-s + 0.635·89-s − 0.419·91-s − 0.205·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.75485818841271, −12.21129560431110, −11.98528015954392, −11.47659258792485, −10.88654665342936, −10.62699608325511, −9.910791989834017, −9.758119096267836, −9.039170027681059, −8.755611688009352, −8.181055407144369, −7.691084363426364, −7.332975099872476, −6.650677075036318, −6.276097651290032, −5.990532419791149, −5.092547595934487, −4.901573640506508, −4.133374122416526, −3.675819680272330, −3.247533962693764, −2.702717715266962, −2.113017942708172, −1.265178846410717, −0.7622793231616612, 0,
0.7622793231616612, 1.265178846410717, 2.113017942708172, 2.702717715266962, 3.247533962693764, 3.675819680272330, 4.133374122416526, 4.901573640506508, 5.092547595934487, 5.990532419791149, 6.276097651290032, 6.650677075036318, 7.332975099872476, 7.691084363426364, 8.181055407144369, 8.755611688009352, 9.039170027681059, 9.758119096267836, 9.910791989834017, 10.62699608325511, 10.88654665342936, 11.47659258792485, 11.98528015954392, 12.21129560431110, 12.75485818841271
