| L(s) = 1 | + 2·3-s + 2·4-s + 5-s + 9-s + 4·12-s + 2·15-s + 2·20-s − 5·23-s − 4·25-s − 4·27-s + 10·31-s + 2·36-s − 14·37-s + 45-s − 7·47-s + 2·49-s + 2·53-s + 9·59-s + 4·60-s − 8·64-s − 18·67-s − 10·69-s − 3·71-s − 8·75-s − 11·81-s + 15·89-s − 10·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4-s + 0.447·5-s + 1/3·9-s + 1.15·12-s + 0.516·15-s + 0.447·20-s − 1.04·23-s − 4/5·25-s − 0.769·27-s + 1.79·31-s + 1/3·36-s − 2.30·37-s + 0.149·45-s − 1.02·47-s + 2/7·49-s + 0.274·53-s + 1.17·59-s + 0.516·60-s − 64-s − 2.19·67-s − 1.20·69-s − 0.356·71-s − 0.923·75-s − 1.22·81-s + 1.58·89-s − 1.04·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.32174396110033485275252370987, −6.91874715076244543084735179757, −6.57196878115659805152469188001, −6.06237513520528007700414794351, −5.81818993572122600709980014977, −5.25604663573377280035369940112, −4.69151093880696795566685281695, −4.21177433863618048450109418602, −3.63266113428945224882450786268, −3.26631167624545286828943213403, −2.70395128959344637055620916381, −2.30030715119460572622255909463, −1.87410963207314380950584746636, −1.36352988561360373004612983076, 0,
1.36352988561360373004612983076, 1.87410963207314380950584746636, 2.30030715119460572622255909463, 2.70395128959344637055620916381, 3.26631167624545286828943213403, 3.63266113428945224882450786268, 4.21177433863618048450109418602, 4.69151093880696795566685281695, 5.25604663573377280035369940112, 5.81818993572122600709980014977, 6.06237513520528007700414794351, 6.57196878115659805152469188001, 6.91874715076244543084735179757, 7.32174396110033485275252370987
