| L(s) = 1 | − 3·7-s − 4·9-s + 4·17-s − 7·23-s − 6·25-s − 4·31-s − 14·47-s + 12·63-s + 2·71-s + 6·73-s − 4·79-s + 7·81-s − 8·89-s + 4·97-s − 6·103-s + 8·113-s − 12·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + 21·161-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 4/3·9-s + 0.970·17-s − 1.45·23-s − 6/5·25-s − 0.718·31-s − 2.04·47-s + 1.51·63-s + 0.237·71-s + 0.702·73-s − 0.450·79-s + 7/9·81-s − 0.847·89-s + 0.406·97-s − 0.591·103-s + 0.752·113-s − 1.10·119-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.29·153-s + 0.0798·157-s + 1.65·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.926145872918632439826198575660, −9.595045959251982254959930061578, −9.037871427034557810662219215244, −8.345575183130610167188007865711, −7.989858284140712077515034529499, −7.45248250543738288777765962737, −6.58979855347549545962881496267, −6.19320850196938924292455976244, −5.67334862307047011095588887286, −5.19310112614586475083982364692, −4.12548383912120958514795534208, −3.46515215889279553519866910129, −2.96590493614543809272616014027, −1.93920488813549103170007233916, 0,
1.93920488813549103170007233916, 2.96590493614543809272616014027, 3.46515215889279553519866910129, 4.12548383912120958514795534208, 5.19310112614586475083982364692, 5.67334862307047011095588887286, 6.19320850196938924292455976244, 6.58979855347549545962881496267, 7.45248250543738288777765962737, 7.989858284140712077515034529499, 8.345575183130610167188007865711, 9.037871427034557810662219215244, 9.595045959251982254959930061578, 9.926145872918632439826198575660
