| L(s) = 1 | − 2-s + 4-s + 4·5-s + 2·7-s − 8-s − 4·10-s + 11-s − 2·14-s + 16-s + 4·20-s − 22-s − 6·23-s + 11·25-s + 2·28-s − 2·29-s − 4·31-s − 32-s + 8·35-s − 2·37-s − 4·40-s − 6·41-s − 4·43-s + 44-s + 6·46-s + 6·47-s − 3·49-s − 11·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.755·7-s − 0.353·8-s − 1.26·10-s + 0.301·11-s − 0.534·14-s + 1/4·16-s + 0.894·20-s − 0.213·22-s − 1.25·23-s + 11/5·25-s + 0.377·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 1.35·35-s − 0.328·37-s − 0.632·40-s − 0.937·41-s − 0.609·43-s + 0.150·44-s + 0.884·46-s + 0.875·47-s − 3/7·49-s − 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.64042062735520, −14.02369251738840, −13.76674694381762, −13.16677559181077, −12.59230408896601, −12.00801033389209, −11.50764274415261, −10.79268983268528, −10.51820399828927, −9.884155161759400, −9.540461855875543, −9.025720591786159, −8.496877912961928, −7.973690828574653, −7.290623847674025, −6.715747225325850, −6.133176297724351, −5.731011377173652, −5.145699993741596, −4.549461172125991, −3.643883800786604, −2.890044124371440, −2.072612154661076, −1.762751010601452, −1.209471165801525, 0,
1.209471165801525, 1.762751010601452, 2.072612154661076, 2.890044124371440, 3.643883800786604, 4.549461172125991, 5.145699993741596, 5.731011377173652, 6.133176297724351, 6.715747225325850, 7.290623847674025, 7.973690828574653, 8.496877912961928, 9.025720591786159, 9.540461855875543, 9.884155161759400, 10.51820399828927, 10.79268983268528, 11.50764274415261, 12.00801033389209, 12.59230408896601, 13.16677559181077, 13.76674694381762, 14.02369251738840, 14.64042062735520
