| L(s) = 1 | + 6·3-s − 4-s − 10·5-s + 27·9-s − 6·12-s − 60·15-s − 63·16-s + 10·20-s + 96·23-s + 75·25-s + 108·27-s + 320·31-s − 27·36-s − 532·37-s − 270·45-s − 1.00e3·47-s − 378·48-s − 446·49-s − 684·53-s − 1.32e3·59-s + 60·60-s + 127·64-s + 992·67-s + 576·69-s − 1.41e3·71-s + 450·75-s + 630·80-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/8·4-s − 0.894·5-s + 9-s − 0.144·12-s − 1.03·15-s − 0.984·16-s + 0.111·20-s + 0.870·23-s + 3/5·25-s + 0.769·27-s + 1.85·31-s − 1/8·36-s − 2.36·37-s − 0.894·45-s − 3.12·47-s − 1.13·48-s − 1.30·49-s − 1.77·53-s − 2.91·59-s + 0.129·60-s + 0.248·64-s + 1.80·67-s + 1.00·69-s − 2.36·71-s + 0.692·75-s + 0.880·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 446 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 466 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9586 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13658 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 5038 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 106102 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 63014 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 504 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 660 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 406922 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 496 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 708 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 364694 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 954338 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1136314 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 606 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
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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
−8.588162672267354936735080070137, −8.377101091443660710738583181545, −8.015174743625218510657658139800, −7.58980685279061000693114662684, −7.30715583388747158873968004724, −6.60918771942042557881963038122, −6.58958265000278309963838252316, −6.14472012216150194959722550708, −5.08734087938487562879417624664, −4.96316912492890356884526983238, −4.54334867391187319414925652661, −4.20648126481112359014036648408, −3.40905520143395471332860680440, −3.11281506120626705124150229800, −3.05049779434701046353107108311, −2.12867523949212963342903612660, −1.62595026429531559448008595461, −1.16494356637207270430693701809, 0, 0,
1.16494356637207270430693701809, 1.62595026429531559448008595461, 2.12867523949212963342903612660, 3.05049779434701046353107108311, 3.11281506120626705124150229800, 3.40905520143395471332860680440, 4.20648126481112359014036648408, 4.54334867391187319414925652661, 4.96316912492890356884526983238, 5.08734087938487562879417624664, 6.14472012216150194959722550708, 6.58958265000278309963838252316, 6.60918771942042557881963038122, 7.30715583388747158873968004724, 7.58980685279061000693114662684, 8.015174743625218510657658139800, 8.377101091443660710738583181545, 8.588162672267354936735080070137
