L(s) = 1 | − 3.17·3-s − 4.23·5-s + 7-s + 7.08·9-s + 4.99·11-s − 0.439·13-s + 13.4·15-s + 3.50·17-s + 3.45·19-s − 3.17·21-s − 4.39·23-s + 12.9·25-s − 12.9·27-s + 0.132·29-s − 5.55·31-s − 15.8·33-s − 4.23·35-s + 0.572·37-s + 1.39·39-s + 5.42·41-s − 5.69·43-s − 30.0·45-s − 9.37·47-s + 49-s − 11.1·51-s + 3.65·53-s − 21.1·55-s + ⋯ |
L(s) = 1 | − 1.83·3-s − 1.89·5-s + 0.377·7-s + 2.36·9-s + 1.50·11-s − 0.121·13-s + 3.47·15-s + 0.849·17-s + 0.791·19-s − 0.693·21-s − 0.916·23-s + 2.59·25-s − 2.49·27-s + 0.0246·29-s − 0.998·31-s − 2.76·33-s − 0.716·35-s + 0.0941·37-s + 0.223·39-s + 0.847·41-s − 0.868·43-s − 4.47·45-s − 1.36·47-s + 0.142·49-s − 1.55·51-s + 0.502·53-s − 2.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6462461404\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6462461404\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 11 | \( 1 - 4.99T + 11T^{2} \) |
| 13 | \( 1 + 0.439T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 - 3.45T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 0.132T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 0.572T + 37T^{2} \) |
| 41 | \( 1 - 5.42T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 - 3.11T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.328882981678545020180399765203, −7.60485384556407751855105154764, −7.04769084280746522889116906931, −6.38531941406020248755342074156, −5.47442770564300043033015478751, −4.75905551851667474551469261887, −4.02426149438878533859477076224, −3.51593816687401860789350400204, −1.44280298636464336005445458616, −0.57478052049921167966830512863,
0.57478052049921167966830512863, 1.44280298636464336005445458616, 3.51593816687401860789350400204, 4.02426149438878533859477076224, 4.75905551851667474551469261887, 5.47442770564300043033015478751, 6.38531941406020248755342074156, 7.04769084280746522889116906931, 7.60485384556407751855105154764, 8.328882981678545020180399765203