| L(s) = 1 | + 2-s + 4-s + 8-s − 4·9-s + 16-s − 4·18-s − 12·29-s + 32-s − 4·36-s + 12·37-s − 7·49-s + 12·53-s − 12·58-s + 64-s − 4·72-s + 12·74-s + 7·81-s − 7·98-s + 12·106-s + 4·109-s − 36·113-s − 12·116-s − 22·121-s + 127-s + 128-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 4/3·9-s + 1/4·16-s − 0.942·18-s − 2.22·29-s + 0.176·32-s − 2/3·36-s + 1.97·37-s − 49-s + 1.64·53-s − 1.57·58-s + 1/8·64-s − 0.471·72-s + 1.39·74-s + 7/9·81-s − 0.707·98-s + 1.16·106-s + 0.383·109-s − 3.38·113-s − 1.11·116-s − 2·121-s + 0.0887·127-s + 0.0883·128-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−7.73980771648409618201814631345, −7.66136888328202171464191211736, −6.90087290692468128106118204132, −6.52232781163114891119640246385, −6.02639889806787953647433445659, −5.53495542743274253939273420459, −5.42709796221295146699028599096, −4.75706789902090315459675700621, −4.10998569722279001202910423614, −3.77138809706715450998803875583, −3.15273308985975549154504171952, −2.59090746251813415753701432976, −2.16673677686876955750453703191, −1.22512445607754846838068936461, 0,
1.22512445607754846838068936461, 2.16673677686876955750453703191, 2.59090746251813415753701432976, 3.15273308985975549154504171952, 3.77138809706715450998803875583, 4.10998569722279001202910423614, 4.75706789902090315459675700621, 5.42709796221295146699028599096, 5.53495542743274253939273420459, 6.02639889806787953647433445659, 6.52232781163114891119640246385, 6.90087290692468128106118204132, 7.66136888328202171464191211736, 7.73980771648409618201814631345
