| L(s) = 1 | − 16-s − 20·25-s + 14·49-s − 16·67-s + 32·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 1/4·16-s − 4·25-s + 2·49-s − 1.95·67-s + 3.60·79-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 + 4·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6036574257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6036574257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 734 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1234 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + 2914 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−11.39760454038416226698064319009, −10.70075553940804794258546210474, −10.63294848724719968717501180714, −10.25628476622249935753522906366, −10.07756726314679396730993129567, −9.633496778371731059670292825705, −9.209824168885707020080376475866, −9.102503204622770221322359963046, −8.964968723886769494525121918646, −8.127330308480759372918817842344, −7.914598166234394637787708660489, −7.86102276266248389000501685141, −7.49854097002460438643128936049, −6.97782560947259423579006266016, −6.51704390189487830140740976882, −6.34971266648369876382104934076, −5.65581644885609870537574479063, −5.64820847737513046977322495616, −5.21610005166815684542841725562, −4.43966950728793303655486985089, −4.14866449520297582156615506030, −3.79668426323100196691206950748, −3.21747895858856631008707763599, −2.36910700508465272032724899456, −1.88132894244762286071273836156,
1.88132894244762286071273836156, 2.36910700508465272032724899456, 3.21747895858856631008707763599, 3.79668426323100196691206950748, 4.14866449520297582156615506030, 4.43966950728793303655486985089, 5.21610005166815684542841725562, 5.64820847737513046977322495616, 5.65581644885609870537574479063, 6.34971266648369876382104934076, 6.51704390189487830140740976882, 6.97782560947259423579006266016, 7.49854097002460438643128936049, 7.86102276266248389000501685141, 7.914598166234394637787708660489, 8.127330308480759372918817842344, 8.964968723886769494525121918646, 9.102503204622770221322359963046, 9.209824168885707020080376475866, 9.633496778371731059670292825705, 10.07756726314679396730993129567, 10.25628476622249935753522906366, 10.63294848724719968717501180714, 10.70075553940804794258546210474, 11.39760454038416226698064319009
