| L(s) = 1 | − 3·9-s − 14·13-s + 10·25-s − 2·37-s − 28·61-s − 14·73-s + 9·81-s + 28·97-s − 34·109-s + 42·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s − 3.88·13-s + 2·25-s − 0.328·37-s − 3.58·61-s − 1.63·73-s + 81-s + 2.84·97-s − 3.25·109-s + 3.88·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1582780393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1582780393\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.057388009487985982741926637639, −9.053035966969722108950545627598, −8.366364291848424234648865922414, −8.000943590282717289561775407830, −7.41954774703763249464139903326, −7.38269028665345657070837334659, −7.07186959440517697353335820752, −6.33533873425176335963614904931, −6.25360107049326898120551386870, −5.45255514103565750986780914797, −5.15218771369243878379535029525, −4.80144729616672893954543188408, −4.65571740471496100887281170939, −4.01026648894047029139493730596, −3.03115392900874834067879813453, −3.03048460560840127825364430557, −2.51263898296052585980890995263, −2.10559207402222168616955538853, −1.24460249660826580575757902951, −0.13523685308600440766328821257,
0.13523685308600440766328821257, 1.24460249660826580575757902951, 2.10559207402222168616955538853, 2.51263898296052585980890995263, 3.03048460560840127825364430557, 3.03115392900874834067879813453, 4.01026648894047029139493730596, 4.65571740471496100887281170939, 4.80144729616672893954543188408, 5.15218771369243878379535029525, 5.45255514103565750986780914797, 6.25360107049326898120551386870, 6.33533873425176335963614904931, 7.07186959440517697353335820752, 7.38269028665345657070837334659, 7.41954774703763249464139903326, 8.000943590282717289561775407830, 8.366364291848424234648865922414, 9.053035966969722108950545627598, 9.057388009487985982741926637639
