| L(s) = 1 | − 4·7-s + 4·13-s + 4·19-s + 4·23-s − 10·29-s + 2·31-s + 2·37-s − 6·41-s − 8·43-s + 9·49-s + 6·53-s + 10·59-s + 2·61-s + 67-s + 10·71-s − 2·73-s − 10·79-s − 8·83-s − 6·89-s − 16·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.10·13-s + 0.917·19-s + 0.834·23-s − 1.85·29-s + 0.359·31-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 9/7·49-s + 0.824·53-s + 1.30·59-s + 0.256·61-s + 0.122·67-s + 1.18·71-s − 0.234·73-s − 1.12·79-s − 0.878·83-s − 0.635·89-s − 1.67·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.16002335057783, −12.84541419907369, −12.23623761851905, −11.63279961487447, −11.33559344030861, −10.86348949377565, −10.15239982559905, −9.883696542222287, −9.477002661779764, −8.894242051729953, −8.577439776711186, −7.987817757575037, −7.164090231299776, −7.099446897302694, −6.409838085060393, −6.049645428556190, −5.388438137251109, −5.165907495339619, −4.151117133896446, −3.767226963042725, −3.276342192817638, −2.915936320275418, −2.127234263278458, −1.398666986833693, −0.7496692425884327, 0,
0.7496692425884327, 1.398666986833693, 2.127234263278458, 2.915936320275418, 3.276342192817638, 3.767226963042725, 4.151117133896446, 5.165907495339619, 5.388438137251109, 6.049645428556190, 6.409838085060393, 7.099446897302694, 7.164090231299776, 7.987817757575037, 8.577439776711186, 8.894242051729953, 9.477002661779764, 9.883696542222287, 10.15239982559905, 10.86348949377565, 11.33559344030861, 11.63279961487447, 12.23623761851905, 12.84541419907369, 13.16002335057783
