| L(s) = 1 | − 5·7-s − 19-s − 5·25-s − 11·31-s + 11·37-s + 18·49-s − 12·61-s − 27·67-s + 3·73-s − 9·79-s + 13·103-s + 19·109-s − 11·121-s + 127-s + 131-s + 5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 25·175-s + 179-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 0.229·19-s − 25-s − 1.97·31-s + 1.80·37-s + 18/7·49-s − 1.53·61-s − 3.29·67-s + 0.351·73-s − 1.01·79-s + 1.28·103-s + 1.81·109-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.433·133-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 − 0.0769·169-s + 0.0760·173-s + 1.88·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7341673472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7341673472\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.13614736388699184611999122108, −9.748234781246634507104937762896, −9.337916803516921899084753846003, −8.984326977143893156218675450510, −8.793676199073621590426559522344, −7.984819082816364742023555531218, −7.54064108802384523592163516580, −7.30486701652472527214542029487, −6.82004046332287558642972975745, −6.12898311810324774963097120620, −6.02201112600387668781992759031, −5.72077218042240509216660539367, −4.92651568058343011153403778636, −4.30296672381606252601703277882, −3.94664402982279755801921646358, −3.29581960159746243522278691003, −2.99855073557405389434607987799, −2.30949448237584298808730273506, −1.56560138545122196908044741465, −0.38458328176277377067320820992,
0.38458328176277377067320820992, 1.56560138545122196908044741465, 2.30949448237584298808730273506, 2.99855073557405389434607987799, 3.29581960159746243522278691003, 3.94664402982279755801921646358, 4.30296672381606252601703277882, 4.92651568058343011153403778636, 5.72077218042240509216660539367, 6.02201112600387668781992759031, 6.12898311810324774963097120620, 6.82004046332287558642972975745, 7.30486701652472527214542029487, 7.54064108802384523592163516580, 7.984819082816364742023555531218, 8.793676199073621590426559522344, 8.984326977143893156218675450510, 9.337916803516921899084753846003, 9.748234781246634507104937762896, 10.13614736388699184611999122108
