| L(s) = 1 | − 2·3-s + 2·7-s + 3·9-s − 4·13-s + 2·19-s − 4·21-s − 8·23-s + 2·25-s − 4·27-s − 8·29-s − 8·31-s − 4·37-s + 8·39-s + 12·41-s − 12·47-s + 3·49-s − 4·57-s − 8·59-s + 12·61-s + 6·63-s − 16·67-s + 16·69-s − 20·71-s + 12·73-s − 4·75-s + 5·81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 9-s − 1.10·13-s + 0.458·19-s − 0.872·21-s − 1.66·23-s + 2/5·25-s − 0.769·27-s − 1.48·29-s − 1.43·31-s − 0.657·37-s + 1.28·39-s + 1.87·41-s − 1.75·47-s + 3/7·49-s − 0.529·57-s − 1.04·59-s + 1.53·61-s + 0.755·63-s − 1.95·67-s + 1.92·69-s − 2.37·71-s + 1.40·73-s − 0.461·75-s + 5/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 230 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.62741239123276330971965320785, −7.50784955470981668478087344720, −7.24353596437252247747510712504, −6.87083343128890967239573089003, −6.24142187524019806197021144680, −6.11666235978260315106507213083, −5.60912840937199909173260299070, −5.46381201951289415562305158847, −4.97815822090921893042469726894, −4.75228485148026595527616933931, −4.23777221874247496611362479275, −4.02052064996590847243994211431, −3.46377178547783112567390327634, −3.05654771172371249414613878305, −2.30678268748838347657966477722, −2.00978967669075797575224498209, −1.58207393914514526105664174470, −1.02796098512217898170726383260, 0, 0,
1.02796098512217898170726383260, 1.58207393914514526105664174470, 2.00978967669075797575224498209, 2.30678268748838347657966477722, 3.05654771172371249414613878305, 3.46377178547783112567390327634, 4.02052064996590847243994211431, 4.23777221874247496611362479275, 4.75228485148026595527616933931, 4.97815822090921893042469726894, 5.46381201951289415562305158847, 5.60912840937199909173260299070, 6.11666235978260315106507213083, 6.24142187524019806197021144680, 6.87083343128890967239573089003, 7.24353596437252247747510712504, 7.50784955470981668478087344720, 7.62741239123276330971965320785
