| L(s) = 1 | − 3·4-s − 6·9-s + 8·11-s + 5·16-s − 5·25-s + 4·29-s + 18·36-s − 24·44-s − 3·64-s + 32·71-s + 16·79-s + 27·81-s − 48·99-s + 15·100-s + 36·109-s − 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 30·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2·9-s + 2.41·11-s + 5/4·16-s − 25-s + 0.742·29-s + 3·36-s − 3.61·44-s − 3/8·64-s + 3.79·71-s + 1.80·79-s + 3·81-s − 4.82·99-s + 3/2·100-s + 3.44·109-s − 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8357738200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8357738200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.540102098470066783229159054034, −9.489525085039792097687572288594, −8.959096240229079857657424888696, −8.498120181782134288657898916471, −8.258764783821744494006777469815, −7.53770562837158976453911852227, −6.47803659589426412986704519993, −6.38920763142189327962156871867, −5.62983515568003563476629870364, −5.08673463814783736975452386140, −4.44056415760512758846670567036, −3.68334690525640900028863674377, −3.45773984941604093394269964405, −2.20922818309632113036387473157, −0.77987991622378332753572806223,
0.77987991622378332753572806223, 2.20922818309632113036387473157, 3.45773984941604093394269964405, 3.68334690525640900028863674377, 4.44056415760512758846670567036, 5.08673463814783736975452386140, 5.62983515568003563476629870364, 6.38920763142189327962156871867, 6.47803659589426412986704519993, 7.53770562837158976453911852227, 8.258764783821744494006777469815, 8.498120181782134288657898916471, 8.959096240229079857657424888696, 9.489525085039792097687572288594, 9.540102098470066783229159054034
