| L(s) = 1 | − 3·5-s + 3·7-s + 7·17-s + 4·19-s + 4·23-s + 4·25-s + 4·29-s + 8·31-s − 9·35-s + 7·37-s − 2·41-s + 43-s − 7·47-s + 2·49-s + 4·53-s + 14·59-s − 10·61-s − 2·67-s + 3·71-s + 14·73-s − 10·79-s + 14·83-s − 21·85-s − 12·95-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s + 1.69·17-s + 0.917·19-s + 0.834·23-s + 4/5·25-s + 0.742·29-s + 1.43·31-s − 1.52·35-s + 1.15·37-s − 0.312·41-s + 0.152·43-s − 1.02·47-s + 2/7·49-s + 0.549·53-s + 1.82·59-s − 1.28·61-s − 0.244·67-s + 0.356·71-s + 1.63·73-s − 1.12·79-s + 1.53·83-s − 2.27·85-s − 1.23·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.685459688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.685459688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.13385750912593, −11.98244061609606, −11.69393374055694, −11.13565314964539, −10.86326653053754, −10.20563748482253, −9.752254062404295, −9.387706218997277, −8.504828230692484, −8.283363838047986, −7.976552041987516, −7.482804659521781, −7.184661864367906, −6.518262306103247, −5.950797023239668, −5.255655162189324, −4.980956455799412, −4.508028616727916, −3.967820356349452, −3.394328231339422, −3.013572135455342, −2.412463323956871, −1.496610230480789, −0.9774375637934310, −0.6222687377474259,
0.6222687377474259, 0.9774375637934310, 1.496610230480789, 2.412463323956871, 3.013572135455342, 3.394328231339422, 3.967820356349452, 4.508028616727916, 4.980956455799412, 5.255655162189324, 5.950797023239668, 6.518262306103247, 7.184661864367906, 7.482804659521781, 7.976552041987516, 8.283363838047986, 8.504828230692484, 9.387706218997277, 9.752254062404295, 10.20563748482253, 10.86326653053754, 11.13565314964539, 11.69393374055694, 11.98244061609606, 12.13385750912593
