| L(s) = 1 | + 6·9-s − 4·25-s + 20·49-s + 56·73-s + 27·81-s + 8·97-s − 20·121-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 − 24·225-s + ⋯ |
| L(s) = 1 | + 2·9-s − 4/5·25-s + 20/7·49-s + 6.55·73-s + 3·81-s + 0.812·97-s − 1.81·121-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 − 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 − 8/5·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.147871534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147871534\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−7.52835756594407838306636187986, −7.23716426122048259564048818696, −6.83324376074904436367170150595, −6.82803893839965645554382810325, −6.48524605960970495130017039276, −6.32327614962902531203610541779, −6.22281190656349411477149969298, −5.73428704221590813478382901402, −5.36615790469085580561597214526, −5.25895141410010010667904006152, −5.19522761895446153394699703063, −4.70513787943132556190126113640, −4.58051703211354682183803819234, −4.06847856522409827648720879197, −4.03924374989051954847480539257, −3.74800611960441040979023191349, −3.69222820900469522900583935906, −3.17149221172108981476098577944, −2.83838686951482260099094827100, −2.27893758085900756219810726486, −2.19664772209953578996182997816, −1.94579469139449183739562668870, −1.41647093101350920286495424003, −0.912602202984102835619506873527, −0.65620606335608006943425025629,
0.65620606335608006943425025629, 0.912602202984102835619506873527, 1.41647093101350920286495424003, 1.94579469139449183739562668870, 2.19664772209953578996182997816, 2.27893758085900756219810726486, 2.83838686951482260099094827100, 3.17149221172108981476098577944, 3.69222820900469522900583935906, 3.74800611960441040979023191349, 4.03924374989051954847480539257, 4.06847856522409827648720879197, 4.58051703211354682183803819234, 4.70513787943132556190126113640, 5.19522761895446153394699703063, 5.25895141410010010667904006152, 5.36615790469085580561597214526, 5.73428704221590813478382901402, 6.22281190656349411477149969298, 6.32327614962902531203610541779, 6.48524605960970495130017039276, 6.82803893839965645554382810325, 6.83324376074904436367170150595, 7.23716426122048259564048818696, 7.52835756594407838306636187986
