| L(s) = 1 | + 5-s − 4·11-s + 13-s − 2·17-s − 2·19-s − 2·23-s + 25-s − 6·29-s + 2·31-s − 2·37-s − 10·41-s − 4·43-s − 8·47-s − 4·55-s − 12·59-s − 8·61-s + 65-s − 4·71-s − 6·73-s + 16·79-s + 4·83-s − 2·85-s − 18·89-s − 2·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s − 0.539·55-s − 1.56·59-s − 1.02·61-s + 0.124·65-s − 0.474·71-s − 0.702·73-s + 1.80·79-s + 0.439·83-s − 0.216·85-s − 1.90·89-s − 0.205·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.28740684465264, −13.04870286278238, −12.59315358222356, −11.96539691148793, −11.57521687797128, −10.96061234298449, −10.54366719591262, −10.28979740709948, −9.636290813887742, −9.254620791223044, −8.687822530733316, −8.154508948238930, −7.861538682465360, −7.252716545913897, −6.567135608408667, −6.375531283982821, −5.587402417540803, −5.302273639511991, −4.702529046254898, −4.231376172137880, −3.438278457655693, −3.050530454683486, −2.358258069606069, −1.821569296944196, −1.323849858573107, 0, 0,
1.323849858573107, 1.821569296944196, 2.358258069606069, 3.050530454683486, 3.438278457655693, 4.231376172137880, 4.702529046254898, 5.302273639511991, 5.587402417540803, 6.375531283982821, 6.567135608408667, 7.252716545913897, 7.861538682465360, 8.154508948238930, 8.687822530733316, 9.254620791223044, 9.636290813887742, 10.28979740709948, 10.54366719591262, 10.96061234298449, 11.57521687797128, 11.96539691148793, 12.59315358222356, 13.04870286278238, 13.28740684465264
