| L(s) = 1 | + 3-s + 4.95·7-s + 9-s − 3.69·11-s − 13-s − 6.38·17-s + 1.25·19-s + 4.95·21-s − 2.64·23-s + 27-s + 5.51·29-s + 9.08·31-s − 3.69·33-s + 6.13·37-s − 39-s + 4.13·41-s + 2.74·43-s + 0.436·47-s + 17.5·49-s − 6.38·51-s + 11.1·53-s + 1.25·57-s + 7.82·59-s − 1.87·61-s + 4.95·63-s + 13.5·67-s − 2.64·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.87·7-s + 0.333·9-s − 1.11·11-s − 0.277·13-s − 1.54·17-s + 0.288·19-s + 1.08·21-s − 0.551·23-s + 0.192·27-s + 1.02·29-s + 1.63·31-s − 0.643·33-s + 1.00·37-s − 0.160·39-s + 0.645·41-s + 0.418·43-s + 0.0636·47-s + 2.50·49-s − 0.894·51-s + 1.52·53-s + 0.166·57-s + 1.01·59-s − 0.239·61-s + 0.623·63-s + 1.66·67-s − 0.318·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.807350874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.807350874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 - 9.08T + 31T^{2} \) |
| 37 | \( 1 - 6.13T + 37T^{2} \) |
| 41 | \( 1 - 4.13T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 - 0.436T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 7.82T + 59T^{2} \) |
| 61 | \( 1 + 1.87T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 9.77T + 73T^{2} \) |
| 79 | \( 1 + 8.03T + 79T^{2} \) |
| 83 | \( 1 - 7.04T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−8.383786428553590850581549732500, −7.88378854883164073403091096451, −7.27783263242604418412515333875, −6.28690897517096058621486064543, −5.25539255378011304124774106272, −4.64874469636663531875803061967, −4.10185514710534561702530070642, −2.55935382006926285289404072965, −2.26969782497205589644648752694, −0.969080094749957779203795261490,
0.969080094749957779203795261490, 2.26969782497205589644648752694, 2.55935382006926285289404072965, 4.10185514710534561702530070642, 4.64874469636663531875803061967, 5.25539255378011304124774106272, 6.28690897517096058621486064543, 7.27783263242604418412515333875, 7.88378854883164073403091096451, 8.383786428553590850581549732500
