| L(s) = 1 | − 160·11-s − 24·19-s + 9.12e3·29-s − 688·31-s − 2.84e4·41-s − 1.79e4·49-s + 7.60e4·59-s − 1.64e4·61-s − 9.69e4·71-s − 1.85e4·79-s − 4.86e4·89-s + 2.10e5·101-s − 8.28e4·109-s − 3.02e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.02e5·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.398·11-s − 0.0152·19-s + 2.01·29-s − 0.128·31-s − 2.64·41-s − 1.07·49-s + 2.84·59-s − 0.564·61-s − 2.28·71-s − 0.334·79-s − 0.650·89-s + 2.05·101-s − 0.668·109-s − 1.88·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.62·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 17998 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 80 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 602042 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2540814 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 12 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8749086 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4560 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 344 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 119558314 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 14240 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 66775178 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 142745586 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 84864886 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 38000 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8206 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2524788838 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48480 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2340933586 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 9264 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6760658886 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 24320 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1515110462 T^{2} + p^{10} T^{4} \) |
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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
−8.971885254172947948982856524211, −8.759990911155048923296774753630, −8.226736693897596627923870266230, −8.151200696081706078717972432446, −7.24219441747613094010903630146, −7.20430956300476274579830956564, −6.48706897365797914736435210875, −6.29427373750045940846621986846, −5.63779247485808698747439641218, −5.14139769746457103471424477164, −4.77125425878262927797985119360, −4.38405931791744144937798433514, −3.46060525933832590087168768057, −3.45241365248239048368801128223, −2.53047997087968762562723473511, −2.32283218809437205607711445841, −1.32466481695184687021147789530, −1.15720636833634770245908925876, 0, 0,
1.15720636833634770245908925876, 1.32466481695184687021147789530, 2.32283218809437205607711445841, 2.53047997087968762562723473511, 3.45241365248239048368801128223, 3.46060525933832590087168768057, 4.38405931791744144937798433514, 4.77125425878262927797985119360, 5.14139769746457103471424477164, 5.63779247485808698747439641218, 6.29427373750045940846621986846, 6.48706897365797914736435210875, 7.20430956300476274579830956564, 7.24219441747613094010903630146, 8.151200696081706078717972432446, 8.226736693897596627923870266230, 8.759990911155048923296774753630, 8.971885254172947948982856524211
