| L(s) = 1 | − 4-s + 2·9-s + 16-s − 4·19-s − 12·29-s − 20·31-s − 2·36-s − 12·41-s + 10·49-s + 12·59-s − 20·61-s − 64-s + 4·76-s + 20·79-s − 5·81-s + 12·89-s + 36·101-s − 28·109-s + 12·116-s − 22·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1/4·16-s − 0.917·19-s − 2.22·29-s − 3.59·31-s − 1/3·36-s − 1.87·41-s + 10/7·49-s + 1.56·59-s − 2.56·61-s − 1/8·64-s + 0.458·76-s + 2.25·79-s − 5/9·81-s + 1.27·89-s + 3.58·101-s − 2.68·109-s + 1.11·116-s − 2·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7772764020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7772764020\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.613282987381695700507569608430, −8.974489503452485386829350196393, −8.870251897658669245079559945803, −8.360412583388371708771774049874, −7.64457002103717533176717913920, −7.48979139091696947488423442883, −7.32800026623228234537959338868, −6.51805210839746098798524053001, −6.45158171289118076398077490958, −5.60196014278191689248707672053, −5.44072145555373755610138290620, −5.09900200147427341179760128723, −4.43145519543671797565592333579, −4.00598051170431795522349557015, −3.53543313057190371931000784845, −3.44519190011522939853774918083, −2.33691944417049711669645687442, −1.90821831536122576289053640466, −1.50113692348340565457121191534, −0.32038122832832343279610805974,
0.32038122832832343279610805974, 1.50113692348340565457121191534, 1.90821831536122576289053640466, 2.33691944417049711669645687442, 3.44519190011522939853774918083, 3.53543313057190371931000784845, 4.00598051170431795522349557015, 4.43145519543671797565592333579, 5.09900200147427341179760128723, 5.44072145555373755610138290620, 5.60196014278191689248707672053, 6.45158171289118076398077490958, 6.51805210839746098798524053001, 7.32800026623228234537959338868, 7.48979139091696947488423442883, 7.64457002103717533176717913920, 8.360412583388371708771774049874, 8.870251897658669245079559945803, 8.974489503452485386829350196393, 9.613282987381695700507569608430
