| L(s) = 1 | − 3-s + 3·4-s + 5-s + 4·7-s + 9-s − 3·12-s − 15-s + 5·16-s + 3·20-s − 4·21-s + 25-s − 27-s + 12·28-s + 4·35-s + 3·36-s − 14·37-s + 2·41-s + 4·43-s + 45-s − 12·47-s − 5·48-s + 9·49-s + 16·59-s − 3·60-s + 4·63-s + 3·64-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3/2·4-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.866·12-s − 0.258·15-s + 5/4·16-s + 0.670·20-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 2.26·28-s + 0.676·35-s + 1/2·36-s − 2.30·37-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 0.721·48-s + 9/7·49-s + 2.08·59-s − 0.387·60-s + 0.503·63-s + 3/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.634884655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.634884655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−9.214265101558607564241599343694, −8.603582719537307636406766430714, −8.197989059802795335671743185661, −7.69327462592131480999754019674, −7.18089026309161485451977735962, −6.77406318795163245812069424582, −6.37249931870022304518705054396, −5.72151871669733249510861539063, −5.18962610685648615755112692444, −4.94464918771264989300516331972, −4.03540026575913304439177032752, −3.34957289736187104920439928781, −2.38688825177143827032623241623, −1.93295514081623797668097658788, −1.25031308135478411843930828914,
1.25031308135478411843930828914, 1.93295514081623797668097658788, 2.38688825177143827032623241623, 3.34957289736187104920439928781, 4.03540026575913304439177032752, 4.94464918771264989300516331972, 5.18962610685648615755112692444, 5.72151871669733249510861539063, 6.37249931870022304518705054396, 6.77406318795163245812069424582, 7.18089026309161485451977735962, 7.69327462592131480999754019674, 8.197989059802795335671743185661, 8.603582719537307636406766430714, 9.214265101558607564241599343694
