| L(s) = 1 | + 3·2-s + 5·4-s − 7·7-s + 3·8-s + 6·11-s − 21·14-s − 11·16-s + 18·22-s − 18·23-s + 25·25-s − 35·28-s + 54·29-s − 45·32-s − 38·37-s + 58·43-s + 30·44-s − 54·46-s + 49·49-s + 75·50-s + 6·53-s − 21·56-s + 162·58-s − 91·64-s − 118·67-s − 114·71-s − 114·74-s − 42·77-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 5/4·4-s − 7-s + 3/8·8-s + 6/11·11-s − 3/2·14-s − 0.687·16-s + 9/11·22-s − 0.782·23-s + 25-s − 5/4·28-s + 1.86·29-s − 1.40·32-s − 1.02·37-s + 1.34·43-s + 0.681·44-s − 1.17·46-s + 49-s + 3/2·50-s + 6/53·53-s − 3/8·56-s + 2.79·58-s − 1.42·64-s − 1.76·67-s − 1.60·71-s − 1.54·74-s − 0.545·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.217886356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217886356\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{2} T^{2} \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 18 T + p^{2} T^{2} \) |
| 29 | \( 1 - 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 + 118 T + p^{2} T^{2} \) |
| 71 | \( 1 + 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.44883267520213941273015926530, −13.65672327732905921175665587594, −12.59790567954738995403667511617, −11.89879685760639900155473694208, −10.38380474582450941736245842462, −8.928976990351916981274361154648, −6.90798668127654392362724394988, −5.92284200893028833752516865048, −4.37436396144114140811611818029, −3.00740241163769420240414042729,
3.00740241163769420240414042729, 4.37436396144114140811611818029, 5.92284200893028833752516865048, 6.90798668127654392362724394988, 8.928976990351916981274361154648, 10.38380474582450941736245842462, 11.89879685760639900155473694208, 12.59790567954738995403667511617, 13.65672327732905921175665587594, 14.44883267520213941273015926530
