| L(s) = 1 | − 3.08·3-s + 4.29·7-s + 6.51·9-s + 1.21·11-s − 5.08·13-s + 2.29·17-s + 19-s − 13.2·21-s + 7.67·23-s − 10.8·27-s − 0.489·29-s − 3.74·33-s + 2·37-s + 15.6·39-s − 4.16·41-s + 12.9·43-s − 5.80·47-s + 11.4·49-s − 7.08·51-s − 1.93·53-s − 3.08·57-s + 11.0·59-s − 5.38·61-s + 27.9·63-s + 2.48·67-s − 23.6·69-s + 11.7·71-s + ⋯ |
| L(s) = 1 | − 1.78·3-s + 1.62·7-s + 2.17·9-s + 0.365·11-s − 1.41·13-s + 0.557·17-s + 0.229·19-s − 2.89·21-s + 1.60·23-s − 2.08·27-s − 0.0909·29-s − 0.651·33-s + 0.328·37-s + 2.51·39-s − 0.650·41-s + 1.97·43-s − 0.847·47-s + 1.63·49-s − 0.991·51-s − 0.266·53-s − 0.408·57-s + 1.44·59-s − 0.688·61-s + 3.52·63-s + 0.304·67-s − 2.85·69-s + 1.39·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391302682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391302682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 + 0.489T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 8.46T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 3.57T + 97T^{2} \) |
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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
−7.55186271723865104678367666235, −7.22607457881191534416496707776, −6.44413213259384277409601927460, −5.57727646473211237410316780747, −4.96884372864441548232976259210, −4.82745281838402131609466740542, −3.86687429543619487850389320636, −2.47186253021691489893408747837, −1.43637794339462311938930335563, −0.72118745540252929047406822590,
0.72118745540252929047406822590, 1.43637794339462311938930335563, 2.47186253021691489893408747837, 3.86687429543619487850389320636, 4.82745281838402131609466740542, 4.96884372864441548232976259210, 5.57727646473211237410316780747, 6.44413213259384277409601927460, 7.22607457881191534416496707776, 7.55186271723865104678367666235
