L(s) = 1 | + 4·3-s + 2·5-s + 6·9-s − 3·11-s − 4·13-s + 8·15-s − 7·17-s + 2·19-s − 7·25-s − 4·27-s − 8·29-s + 8·31-s − 12·33-s + 4·37-s − 16·39-s − 6·41-s − 2·43-s + 12·45-s − 3·47-s − 28·51-s + 2·53-s − 6·55-s + 8·57-s + 8·59-s − 17·61-s − 8·65-s + 6·71-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 0.894·5-s + 2·9-s − 0.904·11-s − 1.10·13-s + 2.06·15-s − 1.69·17-s + 0.458·19-s − 7/5·25-s − 0.769·27-s − 1.48·29-s + 1.43·31-s − 2.08·33-s + 0.657·37-s − 2.56·39-s − 0.937·41-s − 0.304·43-s + 1.78·45-s − 0.437·47-s − 3.92·51-s + 0.274·53-s − 0.809·55-s + 1.05·57-s + 1.04·59-s − 2.17·61-s − 0.992·65-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 15 T + 188 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 125 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 202 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 138 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.71714917823861638993109811044, −7.49511818147467074546082005126, −7.23104876612701806506903569807, −6.90859687520260318448077582540, −6.20283407147173623240099769954, −6.09249779661824202530871607201, −5.55528799533582143705735827223, −5.33064933673659869049511180988, −4.82300820846674955345054979549, −4.35780900733287766682040458910, −4.03452564750507730897093142648, −3.71506135383554409957625003259, −2.97466419594173721666056639922, −2.88681399415882883247649028216, −2.51642645950403880373160898772, −2.25742215999456096568911350087, −1.68847621078451017832376543411, −1.56299965826322635191903234855, 0, 0,
1.56299965826322635191903234855, 1.68847621078451017832376543411, 2.25742215999456096568911350087, 2.51642645950403880373160898772, 2.88681399415882883247649028216, 2.97466419594173721666056639922, 3.71506135383554409957625003259, 4.03452564750507730897093142648, 4.35780900733287766682040458910, 4.82300820846674955345054979549, 5.33064933673659869049511180988, 5.55528799533582143705735827223, 6.09249779661824202530871607201, 6.20283407147173623240099769954, 6.90859687520260318448077582540, 7.23104876612701806506903569807, 7.49511818147467074546082005126, 7.71714917823861638993109811044