| L(s) = 1 | + 3-s + 2.49·5-s − 2.46·7-s + 9-s + 11-s + 13-s + 2.49·15-s + 2.02·17-s − 6.62·19-s − 2.46·21-s + 0.469·23-s + 1.21·25-s + 27-s + 10.4·29-s + 3.66·31-s + 33-s − 6.15·35-s + 4.93·37-s + 39-s + 4.46·41-s + 3.19·43-s + 2.49·45-s + 4.25·47-s − 0.902·49-s + 2.02·51-s + 0.685·53-s + 2.49·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.11·5-s − 0.933·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.643·15-s + 0.490·17-s − 1.51·19-s − 0.538·21-s + 0.0978·23-s + 0.243·25-s + 0.192·27-s + 1.93·29-s + 0.657·31-s + 0.174·33-s − 1.04·35-s + 0.811·37-s + 0.160·39-s + 0.697·41-s + 0.487·43-s + 0.371·45-s + 0.620·47-s − 0.128·49-s + 0.283·51-s + 0.0940·53-s + 0.336·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.759245576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.759245576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 - 0.469T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 3.66T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 - 0.685T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 0.830T + 61T^{2} \) |
| 67 | \( 1 + 0.493T + 67T^{2} \) |
| 71 | \( 1 - 8.01T + 71T^{2} \) |
| 73 | \( 1 - 5.78T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 4.98T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−8.701774756825405244153282284033, −8.021757217231239652639534692531, −6.92434228623305144267093403929, −6.32586096513983611273394424804, −5.85126537071049876142110026898, −4.68088004651073315188684663565, −3.86608972591089108070543241358, −2.83973341198682155659337691105, −2.22409291745404612430374307131, −0.987142032986456971872504186071,
0.987142032986456971872504186071, 2.22409291745404612430374307131, 2.83973341198682155659337691105, 3.86608972591089108070543241358, 4.68088004651073315188684663565, 5.85126537071049876142110026898, 6.32586096513983611273394424804, 6.92434228623305144267093403929, 8.021757217231239652639534692531, 8.701774756825405244153282284033
