| L(s) = 1 | + 2.66·3-s − 1.15·5-s − 2.16·7-s + 4.10·9-s + 11-s + 0.729·13-s − 3.06·15-s − 2.76·17-s − 3.94·19-s − 5.76·21-s − 1.03·23-s − 3.67·25-s + 2.95·27-s + 7.45·29-s − 8.76·31-s + 2.66·33-s + 2.48·35-s − 0.855·37-s + 1.94·39-s + 7.35·41-s + 2.48·43-s − 4.72·45-s − 0.456·47-s − 2.32·49-s − 7.37·51-s + 9.80·53-s − 1.15·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.514·5-s − 0.816·7-s + 1.36·9-s + 0.301·11-s + 0.202·13-s − 0.792·15-s − 0.671·17-s − 0.905·19-s − 1.25·21-s − 0.215·23-s − 0.735·25-s + 0.568·27-s + 1.38·29-s − 1.57·31-s + 0.464·33-s + 0.420·35-s − 0.140·37-s + 0.311·39-s + 1.14·41-s + 0.379·43-s − 0.704·45-s − 0.0666·47-s − 0.332·49-s − 1.03·51-s + 1.34·53-s − 0.155·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 13 | \( 1 - 0.729T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 + 1.03T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 + 8.76T + 31T^{2} \) |
| 37 | \( 1 + 0.855T + 37T^{2} \) |
| 41 | \( 1 - 7.35T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 0.456T + 47T^{2} \) |
| 53 | \( 1 - 9.80T + 53T^{2} \) |
| 59 | \( 1 + 6.36T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−7.82085599319038569508839906524, −7.18337333739824235864012046364, −6.48926788313913926743069661268, −5.69934605306308882250820763008, −4.29766044403372151916141844009, −4.04347439675827770622335680211, −3.15019001122560408161327981484, −2.53463245194287232409460441835, −1.59632252567980659794454884240, 0,
1.59632252567980659794454884240, 2.53463245194287232409460441835, 3.15019001122560408161327981484, 4.04347439675827770622335680211, 4.29766044403372151916141844009, 5.69934605306308882250820763008, 6.48926788313913926743069661268, 7.18337333739824235864012046364, 7.82085599319038569508839906524
