| L(s) = 1 | + 3-s − 1.82·5-s + 1.82·7-s − 2·9-s + 0.828·11-s − 13-s − 1.82·15-s − 4.65·17-s + 4·19-s + 1.82·21-s + 8.82·23-s − 1.65·25-s − 5·27-s + 8.82·29-s + 0.828·33-s − 3.34·35-s − 1.82·37-s − 39-s − 2.82·41-s + 3·43-s + 3.65·45-s − 5.48·47-s − 3.65·49-s − 4.65·51-s + 8.82·53-s − 1.51·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.817·5-s + 0.691·7-s − 0.666·9-s + 0.249·11-s − 0.277·13-s − 0.472·15-s − 1.12·17-s + 0.917·19-s + 0.398·21-s + 1.84·23-s − 0.331·25-s − 0.962·27-s + 1.63·29-s + 0.144·33-s − 0.565·35-s − 0.300·37-s − 0.160·39-s − 0.441·41-s + 0.457·43-s + 0.545·45-s − 0.800·47-s − 0.522·49-s − 0.652·51-s + 1.21·53-s − 0.204·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.928077867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928077867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 3.65T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−8.446612313120232354708047593965, −8.135637171498510029853026471935, −7.17645467254645585160582643841, −6.64446137043552877426986717198, −5.37078700803602352144759085600, −4.79807451060040878123529778354, −3.86012719995105497243860997240, −3.05263449774375084732007761833, −2.19955605842516589021223663149, −0.804854852426770253675824442332,
0.804854852426770253675824442332, 2.19955605842516589021223663149, 3.05263449774375084732007761833, 3.86012719995105497243860997240, 4.79807451060040878123529778354, 5.37078700803602352144759085600, 6.64446137043552877426986717198, 7.17645467254645585160582643841, 8.135637171498510029853026471935, 8.446612313120232354708047593965
