| L(s) = 1 | − 3-s − 5-s + 9-s + 6·11-s + 3·13-s + 15-s − 4·17-s − 19-s − 4·23-s + 25-s − 27-s + 8·29-s + 31-s − 6·33-s − 7·37-s − 3·39-s − 6·41-s − 43-s − 45-s + 2·47-s + 4·51-s − 4·53-s − 6·55-s + 57-s + 8·59-s + 14·61-s − 3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.832·13-s + 0.258·15-s − 0.970·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.179·31-s − 1.04·33-s − 1.15·37-s − 0.480·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s + 0.291·47-s + 0.560·51-s − 0.549·53-s − 0.809·55-s + 0.132·57-s + 1.04·59-s + 1.79·61-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.87071330901794, −14.29418693798345, −13.82770079314085, −13.39109697231123, −12.64364302835582, −12.09694601190197, −11.82802198961007, −11.27394767080686, −10.87791339293702, −10.16736818053562, −9.735834282783745, −8.973931248680771, −8.499137667831511, −8.231457698835370, −7.145127009967053, −6.742248491834562, −6.423522227462599, −5.806255832318278, −5.021502757519502, −4.372082520223618, −3.912124435330891, −3.449483654325270, −2.417093542860978, −1.573534160270856, −0.9959416674135029, 0,
0.9959416674135029, 1.573534160270856, 2.417093542860978, 3.449483654325270, 3.912124435330891, 4.372082520223618, 5.021502757519502, 5.806255832318278, 6.423522227462599, 6.742248491834562, 7.145127009967053, 8.231457698835370, 8.499137667831511, 8.973931248680771, 9.735834282783745, 10.16736818053562, 10.87791339293702, 11.27394767080686, 11.82802198961007, 12.09694601190197, 12.64364302835582, 13.39109697231123, 13.82770079314085, 14.29418693798345, 14.87071330901794
