| L(s) = 1 | + 4·3-s + 4·5-s + 6·9-s − 2·11-s + 16·15-s + 8·23-s + 2·25-s − 4·27-s − 8·33-s + 20·37-s + 24·45-s − 16·47-s + 2·49-s − 12·53-s − 8·55-s + 28·59-s + 20·67-s + 32·69-s + 24·71-s + 8·75-s − 37·81-s − 4·89-s − 4·97-s − 12·99-s − 8·103-s + 80·111-s + 4·113-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s + 2·9-s − 0.603·11-s + 4.13·15-s + 1.66·23-s + 2/5·25-s − 0.769·27-s − 1.39·33-s + 3.28·37-s + 3.57·45-s − 2.33·47-s + 2/7·49-s − 1.64·53-s − 1.07·55-s + 3.64·59-s + 2.44·67-s + 3.85·69-s + 2.84·71-s + 0.923·75-s − 4.11·81-s − 0.423·89-s − 0.406·97-s − 1.20·99-s − 0.788·103-s + 7.59·111-s + 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.523182338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.523182338\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.083867587292064042476745549264, −7.56807245302446849388411409083, −6.78571919296342801026858814862, −6.73107643226946083552289919888, −5.98411845567390687759228409904, −5.64130249043000575312616744005, −5.20699803882837226838184764640, −4.74417671255472056386371788974, −3.95581538224482392108768250754, −3.56276783709589056726776122484, −3.01367372185754308448342171488, −2.46608091328452001690059553408, −2.36895617399553888783865863574, −1.85344569828924861759779421426, −0.977853460820428219258604150547,
0.977853460820428219258604150547, 1.85344569828924861759779421426, 2.36895617399553888783865863574, 2.46608091328452001690059553408, 3.01367372185754308448342171488, 3.56276783709589056726776122484, 3.95581538224482392108768250754, 4.74417671255472056386371788974, 5.20699803882837226838184764640, 5.64130249043000575312616744005, 5.98411845567390687759228409904, 6.73107643226946083552289919888, 6.78571919296342801026858814862, 7.56807245302446849388411409083, 8.083867587292064042476745549264
