| L(s) = 1 | − 4.60·2-s + 3·3-s + 13.1·4-s − 13.8·6-s + 7·7-s − 23.8·8-s + 9·9-s − 52.9·11-s + 39.5·12-s + 19.6·13-s − 32.2·14-s + 4.19·16-s + 61.5·17-s − 41.4·18-s + 27.0·19-s + 21·21-s + 243.·22-s + 19.2·23-s − 71.4·24-s − 90.2·26-s + 27·27-s + 92.2·28-s − 167.·29-s + 225.·31-s + 171.·32-s − 158.·33-s − 283.·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.577·3-s + 1.64·4-s − 0.939·6-s + 0.377·7-s − 1.05·8-s + 0.333·9-s − 1.45·11-s + 0.950·12-s + 0.418·13-s − 0.614·14-s + 0.0655·16-s + 0.877·17-s − 0.542·18-s + 0.327·19-s + 0.218·21-s + 2.36·22-s + 0.174·23-s − 0.607·24-s − 0.680·26-s + 0.192·27-s + 0.622·28-s − 1.07·29-s + 1.30·31-s + 0.945·32-s − 0.837·33-s − 1.42·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.043540798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043540798\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 4.60T + 8T^{2} \) |
| 11 | \( 1 + 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 311.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 12.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 114.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 207.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 227.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 605.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 315.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 720.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 56.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 692.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 661.T + 9.12e5T^{2} \) |
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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
−10.21601221776570123904641033503, −9.546882571082604506053528356760, −8.617061051234611303854972182588, −7.84470029058768427017118257787, −7.50179620900643305834993893748, −6.12431002378088758982728224238, −4.81634607216087304675965402979, −3.14319674661012380147651949627, −2.04656329438411289135604516172, −0.78400160954035339870385112698,
0.78400160954035339870385112698, 2.04656329438411289135604516172, 3.14319674661012380147651949627, 4.81634607216087304675965402979, 6.12431002378088758982728224238, 7.50179620900643305834993893748, 7.84470029058768427017118257787, 8.617061051234611303854972182588, 9.546882571082604506053528356760, 10.21601221776570123904641033503
