| L(s) = 1 | − 2·5-s − 4·7-s − 11-s + 17-s + 3·19-s + 8·23-s − 25-s + 5·29-s + 2·31-s + 8·35-s + 6·37-s + 3·41-s − 11·43-s + 2·47-s + 9·49-s − 12·53-s + 2·55-s − 2·59-s − 8·61-s − 7·67-s + 8·71-s + 4·73-s + 4·77-s − 10·79-s + 6·83-s − 2·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.301·11-s + 0.242·17-s + 0.688·19-s + 1.66·23-s − 1/5·25-s + 0.928·29-s + 0.359·31-s + 1.35·35-s + 0.986·37-s + 0.468·41-s − 1.67·43-s + 0.291·47-s + 9/7·49-s − 1.64·53-s + 0.269·55-s − 0.260·59-s − 1.02·61-s − 0.855·67-s + 0.949·71-s + 0.468·73-s + 0.455·77-s − 1.12·79-s + 0.658·83-s − 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.81963982104613, −13.45921332359443, −12.77400924929024, −12.72300258779229, −11.94840200706077, −11.65882115382854, −11.02660799016772, −10.55744695777945, −9.939371885131566, −9.568584797503604, −9.093876466679915, −8.504515968729625, −7.891987678715868, −7.462137643895979, −6.978743798093645, −6.286717138949322, −6.127662541813103, −5.091742479871928, −4.836566864946190, −4.017029460724087, −3.451777023526650, −2.993710645391235, −2.641682179684229, −1.448119621076965, −0.7153927441027272, 0,
0.7153927441027272, 1.448119621076965, 2.641682179684229, 2.993710645391235, 3.451777023526650, 4.017029460724087, 4.836566864946190, 5.091742479871928, 6.127662541813103, 6.286717138949322, 6.978743798093645, 7.462137643895979, 7.891987678715868, 8.504515968729625, 9.093876466679915, 9.568584797503604, 9.939371885131566, 10.55744695777945, 11.02660799016772, 11.65882115382854, 11.94840200706077, 12.72300258779229, 12.77400924929024, 13.45921332359443, 13.81963982104613
