| L(s) = 1 | − 33.0·7-s − 48.3·11-s − 60.3·13-s + 17.7·17-s − 130.·19-s − 70.8·23-s − 104.·29-s − 210.·31-s + 300.·37-s − 240.·41-s − 108·43-s + 278.·47-s + 750.·49-s − 328.·53-s − 889.·59-s − 241.·61-s + 103.·67-s + 277.·71-s + 274.·73-s + 1.59e3·77-s + 366.·79-s − 57.7·83-s + 203.·89-s + 1.99e3·91-s + 1.28e3·97-s − 886.·101-s − 783.·103-s + ⋯ |
| L(s) = 1 | − 1.78·7-s − 1.32·11-s − 1.28·13-s + 0.253·17-s − 1.58·19-s − 0.642·23-s − 0.669·29-s − 1.21·31-s + 1.33·37-s − 0.914·41-s − 0.383·43-s + 0.865·47-s + 2.18·49-s − 0.851·53-s − 1.96·59-s − 0.506·61-s + 0.189·67-s + 0.464·71-s + 0.439·73-s + 2.36·77-s + 0.522·79-s − 0.0763·83-s + 0.241·89-s + 2.29·91-s + 1.34·97-s − 0.873·101-s − 0.749·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08512353397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08512353397\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 33.0T + 343T^{2} \) |
| 11 | \( 1 + 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 240.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108T + 7.95e4T^{2} \) |
| 47 | \( 1 - 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 889.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 241.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 277.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 274.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 366.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 57.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 203.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−9.124833632226632554576929251784, −7.997083928626377568364561291898, −7.35860093414685745382703628278, −6.47628735740107843876610425498, −5.81195302182022836176206653888, −4.85960270070861601852114090250, −3.80850551897707933163986641038, −2.86946076378990649949009203084, −2.14841320790883430608661676800, −0.12479841809012241260440945060,
0.12479841809012241260440945060, 2.14841320790883430608661676800, 2.86946076378990649949009203084, 3.80850551897707933163986641038, 4.85960270070861601852114090250, 5.81195302182022836176206653888, 6.47628735740107843876610425498, 7.35860093414685745382703628278, 7.997083928626377568364561291898, 9.124833632226632554576929251784
