| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 5·13-s + 6·19-s − 6·21-s − 23-s − 5·25-s − 9·27-s − 9·29-s − 3·31-s + 8·37-s + 15·39-s − 3·41-s − 8·43-s + 7·47-s − 3·49-s − 2·53-s − 18·57-s + 4·59-s + 10·61-s + 12·63-s + 8·67-s + 3·69-s − 7·71-s − 9·73-s + 15·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 1.38·13-s + 1.37·19-s − 1.30·21-s − 0.208·23-s − 25-s − 1.73·27-s − 1.67·29-s − 0.538·31-s + 1.31·37-s + 2.40·39-s − 0.468·41-s − 1.21·43-s + 1.02·47-s − 3/7·49-s − 0.274·53-s − 2.38·57-s + 0.520·59-s + 1.28·61-s + 1.51·63-s + 0.977·67-s + 0.361·69-s − 0.830·71-s − 1.05·73-s + 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.63482345386735, −14.41743934081811, −13.47806086018703, −13.09495526222172, −12.54122965335065, −11.85641490440086, −11.65379384488716, −11.36471711523928, −10.69597663630913, −10.13560631329651, −9.627788618060958, −9.338576695329187, −8.247117632228316, −7.696928081413374, −7.245999274887558, −6.851789030105918, −5.969373737680227, −5.522566812234913, −5.217103556644284, −4.604437007196599, −4.062947213585378, −3.251989435623953, −2.180339848053201, −1.644345366226787, −0.7563677578747864, 0,
0.7563677578747864, 1.644345366226787, 2.180339848053201, 3.251989435623953, 4.062947213585378, 4.604437007196599, 5.217103556644284, 5.522566812234913, 5.969373737680227, 6.851789030105918, 7.245999274887558, 7.696928081413374, 8.247117632228316, 9.338576695329187, 9.627788618060958, 10.13560631329651, 10.69597663630913, 11.36471711523928, 11.65379384488716, 11.85641490440086, 12.54122965335065, 13.09495526222172, 13.47806086018703, 14.41743934081811, 14.63482345386735
