| L(s) = 1 | + 16·13-s + 28·19-s − 2·25-s − 8·43-s + 2·49-s + 32·67-s + 24·103-s + 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 4.43·13-s + 6.42·19-s − 2/5·25-s − 1.21·43-s + 2/7·49-s + 3.90·67-s + 2.36·103-s + 3.81·121-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 + 8.30·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.54559970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.54559970\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 93 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.18309673051646118861449868317, −6.00710067362678041573220941491, −5.59758593285001821120463490856, −5.51877952544068842863396972827, −5.36002050771621497614467552163, −5.29407275213969366936365073209, −5.20413713847180051637261087870, −4.79764974446538236953512707415, −4.56790246824092573342639219723, −4.15511853916350327083170011893, −3.98747243287054528475596738222, −3.74003356393194628110889943261, −3.67229092852187058632149825292, −3.26147689689149647557610023864, −3.16892381833001943552503352834, −3.16638824158979440098255821801, −3.15922733100283027694971670124, −2.38003464028532640244395000210, −2.25047014565339984894246698994, −1.74607421110152453045737987739, −1.52511870327712413696911063719, −1.14859618952979730486750268422, −1.13005992706493845360334356819, −0.830642958132216915661956436200, −0.64915745649792418934194978111,
0.64915745649792418934194978111, 0.830642958132216915661956436200, 1.13005992706493845360334356819, 1.14859618952979730486750268422, 1.52511870327712413696911063719, 1.74607421110152453045737987739, 2.25047014565339984894246698994, 2.38003464028532640244395000210, 3.15922733100283027694971670124, 3.16638824158979440098255821801, 3.16892381833001943552503352834, 3.26147689689149647557610023864, 3.67229092852187058632149825292, 3.74003356393194628110889943261, 3.98747243287054528475596738222, 4.15511853916350327083170011893, 4.56790246824092573342639219723, 4.79764974446538236953512707415, 5.20413713847180051637261087870, 5.29407275213969366936365073209, 5.36002050771621497614467552163, 5.51877952544068842863396972827, 5.59758593285001821120463490856, 6.00710067362678041573220941491, 6.18309673051646118861449868317
