| L(s) = 1 | + 0.829·2-s + 2.15·3-s − 1.31·4-s + 5-s + 1.78·6-s − 1.04·7-s − 2.74·8-s + 1.63·9-s + 0.829·10-s + 11-s − 2.82·12-s − 4.99·13-s − 0.865·14-s + 2.15·15-s + 0.341·16-s + 2.47·17-s + 1.35·18-s + 2.20·19-s − 1.31·20-s − 2.24·21-s + 0.829·22-s + 5.73·23-s − 5.91·24-s + 25-s − 4.14·26-s − 2.94·27-s + 1.36·28-s + ⋯ |
| L(s) = 1 | + 0.586·2-s + 1.24·3-s − 0.655·4-s + 0.447·5-s + 0.729·6-s − 0.394·7-s − 0.971·8-s + 0.544·9-s + 0.262·10-s + 0.301·11-s − 0.814·12-s − 1.38·13-s − 0.231·14-s + 0.555·15-s + 0.0854·16-s + 0.601·17-s + 0.319·18-s + 0.506·19-s − 0.293·20-s − 0.489·21-s + 0.176·22-s + 1.19·23-s − 1.20·24-s + 0.200·25-s − 0.813·26-s − 0.565·27-s + 0.258·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.326714339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.326714339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 0.829T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 13 | \( 1 + 4.99T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 + 4.99T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 1.05T + 61T^{2} \) |
| 67 | \( 1 - 7.00T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−8.433581757954416421770310839337, −7.936260476473249371071600979851, −6.94793359188713571904188417555, −6.23913667574727684641283547274, −5.17166655711424069392820367564, −4.73213487996104821164134225991, −3.69085522242683386270857908058, −2.95130617047330262038183769421, −2.48487168498962943018554784147, −0.916351500627204995195981911081,
0.916351500627204995195981911081, 2.48487168498962943018554784147, 2.95130617047330262038183769421, 3.69085522242683386270857908058, 4.73213487996104821164134225991, 5.17166655711424069392820367564, 6.23913667574727684641283547274, 6.94793359188713571904188417555, 7.936260476473249371071600979851, 8.433581757954416421770310839337
