| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 11-s + 8·15-s + 16·23-s + 2·25-s + 4·27-s + 2·33-s − 12·37-s + 12·45-s − 10·49-s + 28·53-s + 4·55-s + 24·59-s − 8·67-s + 32·69-s + 4·75-s + 5·81-s − 28·89-s − 4·97-s + 3·99-s + 24·103-s − 24·111-s − 20·113-s + 64·115-s + 121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 0.301·11-s + 2.06·15-s + 3.33·23-s + 2/5·25-s + 0.769·27-s + 0.348·33-s − 1.97·37-s + 1.78·45-s − 1.42·49-s + 3.84·53-s + 0.539·55-s + 3.12·59-s − 0.977·67-s + 3.85·69-s + 0.461·75-s + 5/9·81-s − 2.96·89-s − 0.406·97-s + 0.301·99-s + 2.36·103-s − 2.27·111-s − 1.88·113-s + 5.96·115-s + 1/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3066624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3066624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.568920968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.568920968\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−7.26983803440929507346556269687, −7.02339560430258550540873229128, −7.01506907872700944037082793635, −6.38383875473402815541456430546, −5.78263471331879123598345638239, −5.34474711434827294650147398392, −5.22340952757125710050854485601, −4.60150046530904098097171113225, −3.90041657716003839041565024444, −3.57524581761396247236454027882, −2.91854564551489641468247090209, −2.55394013627242147469753340699, −2.07502595153963756150674220791, −1.49985313315727000952810842633, −0.965181733530667589348302449434,
0.965181733530667589348302449434, 1.49985313315727000952810842633, 2.07502595153963756150674220791, 2.55394013627242147469753340699, 2.91854564551489641468247090209, 3.57524581761396247236454027882, 3.90041657716003839041565024444, 4.60150046530904098097171113225, 5.22340952757125710050854485601, 5.34474711434827294650147398392, 5.78263471331879123598345638239, 6.38383875473402815541456430546, 7.01506907872700944037082793635, 7.02339560430258550540873229128, 7.26983803440929507346556269687
