| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 7-s + 4·10-s − 11-s − 2·14-s − 4·16-s − 2·17-s + 6·19-s − 4·20-s + 2·22-s + 2·23-s − 25-s + 2·28-s + 9·29-s − 9·31-s + 8·32-s + 4·34-s − 2·35-s + 4·37-s − 12·38-s + 3·41-s + 2·43-s − 2·44-s − 4·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 0.377·7-s + 1.26·10-s − 0.301·11-s − 0.534·14-s − 16-s − 0.485·17-s + 1.37·19-s − 0.894·20-s + 0.426·22-s + 0.417·23-s − 1/5·25-s + 0.377·28-s + 1.67·29-s − 1.61·31-s + 1.41·32-s + 0.685·34-s − 0.338·35-s + 0.657·37-s − 1.94·38-s + 0.468·41-s + 0.304·43-s − 0.301·44-s − 0.589·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.86758492172510, −13.31414698719048, −12.83968097580709, −12.11423731087897, −11.75382223418545, −11.13743701079731, −11.05754373021432, −10.33933968497594, −9.851326685936572, −9.431202741847490, −8.798989584825910, −8.522849050117497, −7.840477206275542, −7.608499799163148, −7.177630703607626, −6.594099784208858, −5.894435462744885, −5.172044333007488, −4.628657811801528, −4.174282142965459, −3.329983396123625, −2.839028485730755, −2.017814720211202, −1.365622231851804, −0.7172746585856833, 0,
0.7172746585856833, 1.365622231851804, 2.017814720211202, 2.839028485730755, 3.329983396123625, 4.174282142965459, 4.628657811801528, 5.172044333007488, 5.894435462744885, 6.594099784208858, 7.177630703607626, 7.608499799163148, 7.840477206275542, 8.522849050117497, 8.798989584825910, 9.431202741847490, 9.851326685936572, 10.33933968497594, 11.05754373021432, 11.13743701079731, 11.75382223418545, 12.11423731087897, 12.83968097580709, 13.31414698719048, 13.86758492172510
