| L(s) = 1 | − 2·4-s + 8·7-s + 3·16-s − 14·25-s − 16·28-s + 28·37-s − 8·43-s + 34·49-s − 4·64-s + 8·67-s − 40·79-s + 28·100-s + 52·109-s + 24·112-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 56·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 16·172-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s + 3/4·16-s − 2.79·25-s − 3.02·28-s + 4.60·37-s − 1.21·43-s + 34/7·49-s − 1/2·64-s + 0.977·67-s − 4.50·79-s + 14/5·100-s + 4.98·109-s + 2.26·112-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.60·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 1.21·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330143512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330143512\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.248020643847419198410929074308, −7.88175738467019999506348158549, −7.81229356765347982834276473699, −7.65552021206789772386952219897, −7.54394187372462970620798312318, −7.13760509092754352747438475199, −6.84991170379380529438962191151, −6.12703243987095352421551891844, −6.12390283099408262641506106817, −6.04563459151078904046672920376, −5.52125640432607666596241530268, −5.45834006401551436258061214975, −4.93411973703737854504041244834, −4.75680924392216707947067771664, −4.70629313770613856950238198729, −4.15556038138719123885405847202, −4.08282557117265537752476112023, −3.94386026353510527376359751277, −3.41879298526214755288133422425, −2.71904355305484860998769327284, −2.58436935133502546713499710304, −2.09103980985007047506957425150, −1.51008882502341006757253193254, −1.48579618099667282894445045004, −0.65742208539132196488344514468,
0.65742208539132196488344514468, 1.48579618099667282894445045004, 1.51008882502341006757253193254, 2.09103980985007047506957425150, 2.58436935133502546713499710304, 2.71904355305484860998769327284, 3.41879298526214755288133422425, 3.94386026353510527376359751277, 4.08282557117265537752476112023, 4.15556038138719123885405847202, 4.70629313770613856950238198729, 4.75680924392216707947067771664, 4.93411973703737854504041244834, 5.45834006401551436258061214975, 5.52125640432607666596241530268, 6.04563459151078904046672920376, 6.12390283099408262641506106817, 6.12703243987095352421551891844, 6.84991170379380529438962191151, 7.13760509092754352747438475199, 7.54394187372462970620798312318, 7.65552021206789772386952219897, 7.81229356765347982834276473699, 7.88175738467019999506348158549, 8.248020643847419198410929074308
