| L(s) = 1 | + 3-s + 5-s + 9-s − 8·11-s + 15-s − 12·17-s + 25-s + 27-s − 8·33-s + 24·43-s + 45-s − 14·49-s − 12·51-s + 12·53-s − 8·55-s + 24·59-s + 28·61-s + 8·67-s + 16·71-s + 75-s + 81-s − 12·85-s − 8·99-s − 36·109-s − 12·113-s + 26·121-s + 125-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 2.41·11-s + 0.258·15-s − 2.91·17-s + 1/5·25-s + 0.192·27-s − 1.39·33-s + 3.65·43-s + 0.149·45-s − 2·49-s − 1.68·51-s + 1.64·53-s − 1.07·55-s + 3.12·59-s + 3.58·61-s + 0.977·67-s + 1.89·71-s + 0.115·75-s + 1/9·81-s − 1.30·85-s − 0.804·99-s − 3.44·109-s − 1.12·113-s + 2.36·121-s + 0.0894·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664469560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664469560\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.391564530969274177614715751500, −8.323057177226078640880314002796, −7.84433918506098122901785888900, −7.10648850833945378873095434686, −6.89270810769313104844169224865, −6.43925783086057872365275625126, −5.59585158632682227968586342616, −5.31059979041689708302017443740, −4.90112278095406188180879238277, −4.01130829705839256794389128825, −3.95303834850677407956821293662, −2.63672797798552915147078140108, −2.38987395166756440433579405452, −2.25002432004047770108316173263, −0.63962793409840065289934962638,
0.63962793409840065289934962638, 2.25002432004047770108316173263, 2.38987395166756440433579405452, 2.63672797798552915147078140108, 3.95303834850677407956821293662, 4.01130829705839256794389128825, 4.90112278095406188180879238277, 5.31059979041689708302017443740, 5.59585158632682227968586342616, 6.43925783086057872365275625126, 6.89270810769313104844169224865, 7.10648850833945378873095434686, 7.84433918506098122901785888900, 8.323057177226078640880314002796, 8.391564530969274177614715751500
