| L(s) = 1 | − 1.38·2-s + 3-s − 0.0917·4-s + 5-s − 1.38·6-s − 4.36·7-s + 2.88·8-s + 9-s − 1.38·10-s − 4.31·11-s − 0.0917·12-s + 6.36·13-s + 6.02·14-s + 15-s − 3.80·16-s + 5.71·17-s − 1.38·18-s − 0.0917·20-s − 4.36·21-s + 5.96·22-s − 0.579·23-s + 2.88·24-s + 25-s − 8.78·26-s + 27-s + 0.400·28-s + 3.55·29-s + ⋯ |
| L(s) = 1 | − 0.976·2-s + 0.577·3-s − 0.0458·4-s + 0.447·5-s − 0.563·6-s − 1.64·7-s + 1.02·8-s + 0.333·9-s − 0.436·10-s − 1.30·11-s − 0.0264·12-s + 1.76·13-s + 1.61·14-s + 0.258·15-s − 0.952·16-s + 1.38·17-s − 0.325·18-s − 0.0205·20-s − 0.952·21-s + 1.27·22-s − 0.120·23-s + 0.589·24-s + 0.200·25-s − 1.72·26-s + 0.192·27-s + 0.0756·28-s + 0.659·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080308663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080308663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 23 | \( 1 + 0.579T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 - 0.762T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 4.00T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 - 8.08T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.67T + 83T^{2} \) |
| 89 | \( 1 + 9.84T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.189993162056730788438087356045, −7.80410151484902236201139234374, −6.86451183725154980735736697003, −6.13457011955478612680161010366, −5.47702193755536052183395048526, −4.36892577373737734072999174520, −3.36435731162544093966857137855, −2.93823951013180700871385913908, −1.66057061958957648926926640924, −0.64859685443543141162276940534,
0.64859685443543141162276940534, 1.66057061958957648926926640924, 2.93823951013180700871385913908, 3.36435731162544093966857137855, 4.36892577373737734072999174520, 5.47702193755536052183395048526, 6.13457011955478612680161010366, 6.86451183725154980735736697003, 7.80410151484902236201139234374, 8.189993162056730788438087356045
