| L(s) = 1 | − 24·23-s − 2·25-s − 24·47-s − 4·49-s − 48·71-s + 8·73-s − 56·97-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 + ⋯ |
| L(s) = 1 | − 5.00·23-s − 2/5·25-s − 3.50·47-s − 4/7·49-s − 5.69·71-s + 0.936·73-s − 5.68·97-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5459878561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5459878561\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−6.11118046156431210235715510093, −5.99897481497844312056591823459, −5.97451802258817382971084136297, −5.52947927953441364970317853856, −5.47891643217974240133198250598, −5.41631351465746174666060196957, −4.94467238443910513140050693629, −4.56407856206862881195630493535, −4.54307534668246995040752078356, −4.43128447502564095531935717681, −4.10786555906113252774288593088, −4.00165104397947518667006336784, −3.64924711182703012214796736622, −3.53893830309588698334338595485, −3.30683945468763352187193531920, −2.83180644050823276187235041427, −2.82577109003552860306307379610, −2.52808784570036651926924915127, −2.18984843017715425446545315837, −1.71513017370346432010615460703, −1.68152470817381540955897083669, −1.63564442881427955586536715276, −1.24674177743422189053078997279, −0.35039271820033130796418310580, −0.20632366111293867949882281991,
0.20632366111293867949882281991, 0.35039271820033130796418310580, 1.24674177743422189053078997279, 1.63564442881427955586536715276, 1.68152470817381540955897083669, 1.71513017370346432010615460703, 2.18984843017715425446545315837, 2.52808784570036651926924915127, 2.82577109003552860306307379610, 2.83180644050823276187235041427, 3.30683945468763352187193531920, 3.53893830309588698334338595485, 3.64924711182703012214796736622, 4.00165104397947518667006336784, 4.10786555906113252774288593088, 4.43128447502564095531935717681, 4.54307534668246995040752078356, 4.56407856206862881195630493535, 4.94467238443910513140050693629, 5.41631351465746174666060196957, 5.47891643217974240133198250598, 5.52947927953441364970317853856, 5.97451802258817382971084136297, 5.99897481497844312056591823459, 6.11118046156431210235715510093
