| L(s) = 1 | − 0.871·2-s − 3.06·3-s − 1.24·4-s + 3.32·5-s + 2.66·6-s − 2.06·7-s + 2.82·8-s + 6.39·9-s − 2.89·10-s − 11-s + 3.80·12-s + 1.79·14-s − 10.1·15-s + 0.0224·16-s + 1.15·17-s − 5.56·18-s + 4.63·19-s − 4.12·20-s + 6.32·21-s + 0.871·22-s + 6.38·23-s − 8.65·24-s + 6.06·25-s − 10.3·27-s + 2.56·28-s + 3.75·29-s + 8.88·30-s + ⋯ |
| L(s) = 1 | − 0.616·2-s − 1.76·3-s − 0.620·4-s + 1.48·5-s + 1.08·6-s − 0.780·7-s + 0.998·8-s + 2.13·9-s − 0.916·10-s − 0.301·11-s + 1.09·12-s + 0.480·14-s − 2.63·15-s + 0.00561·16-s + 0.280·17-s − 1.31·18-s + 1.06·19-s − 0.923·20-s + 1.38·21-s + 0.185·22-s + 1.33·23-s − 1.76·24-s + 1.21·25-s − 2.00·27-s + 0.484·28-s + 0.697·29-s + 1.62·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6380611283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6380611283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.871T + 2T^{2} \) |
| 3 | \( 1 + 3.06T + 3T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 - 0.727T + 37T^{2} \) |
| 41 | \( 1 + 0.908T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 0.353T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 8.01T + 71T^{2} \) |
| 73 | \( 1 - 2.09T + 73T^{2} \) |
| 79 | \( 1 + 1.62T + 79T^{2} \) |
| 83 | \( 1 - 8.35T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 1.80T + 97T^{2} \) |
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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
−9.635708492622176783179870554241, −8.757724872761822255297324273206, −7.42965367510506120961938367289, −6.73292373574841754762268399486, −5.96414979662492194214692189230, −5.23944071140075318062808776182, −4.85815230909321599322927298662, −3.35135748919387977943987320871, −1.67423679797966075355041775246, −0.68683224348634562025056212640,
0.68683224348634562025056212640, 1.67423679797966075355041775246, 3.35135748919387977943987320871, 4.85815230909321599322927298662, 5.23944071140075318062808776182, 5.96414979662492194214692189230, 6.73292373574841754762268399486, 7.42965367510506120961938367289, 8.757724872761822255297324273206, 9.635708492622176783179870554241
