| L(s) = 1 | − 2·3-s + 3·9-s − 16·17-s − 4·23-s + 6·25-s − 4·27-s + 20·29-s + 14·43-s + 13·49-s + 32·51-s − 10·61-s + 8·69-s − 12·75-s + 2·79-s + 5·81-s − 40·87-s − 12·101-s + 38·103-s − 4·107-s + 12·113-s − 14·121-s + 127-s − 28·129-s + 131-s + 137-s + 139-s − 26·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 3.88·17-s − 0.834·23-s + 6/5·25-s − 0.769·27-s + 3.71·29-s + 2.13·43-s + 13/7·49-s + 4.48·51-s − 1.28·61-s + 0.963·69-s − 1.38·75-s + 0.225·79-s + 5/9·81-s − 4.28·87-s − 1.19·101-s + 3.74·103-s − 0.386·107-s + 1.12·113-s − 1.27·121-s + 0.0887·127-s − 2.46·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.14·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.521279657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521279657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.565804507121894637890671930752, −8.509268216914703564366598316261, −7.88206089237213746841884978011, −7.25410496408505476935383567816, −7.01850251779180950308209177687, −6.85914857581899063188256516543, −6.19610178802487467559803328133, −6.12826673871206710959436766489, −6.01154439654539057371246227277, −5.02727695367717622545208443219, −4.79782717530575949507958014033, −4.62501020924450705010188838556, −4.16462311462510877599127445507, −3.98089605523346341482641366546, −3.01350569639819847185699685948, −2.59659461810857331554348584680, −2.29322162616000058867333598524, −1.70865554425335682827068743298, −0.803718659270852107838485793607, −0.52935854570143225604337182426,
0.52935854570143225604337182426, 0.803718659270852107838485793607, 1.70865554425335682827068743298, 2.29322162616000058867333598524, 2.59659461810857331554348584680, 3.01350569639819847185699685948, 3.98089605523346341482641366546, 4.16462311462510877599127445507, 4.62501020924450705010188838556, 4.79782717530575949507958014033, 5.02727695367717622545208443219, 6.01154439654539057371246227277, 6.12826673871206710959436766489, 6.19610178802487467559803328133, 6.85914857581899063188256516543, 7.01850251779180950308209177687, 7.25410496408505476935383567816, 7.88206089237213746841884978011, 8.509268216914703564366598316261, 8.565804507121894637890671930752
