| L(s) = 1 | − 3-s + 9-s − 4·13-s + 8·23-s − 2·25-s − 27-s + 4·37-s + 4·39-s + 8·47-s + 6·49-s + 8·59-s + 4·61-s − 8·69-s + 8·71-s − 4·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s − 8·107-s − 20·109-s − 4·111-s − 4·117-s − 22·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.66·23-s − 2/5·25-s − 0.192·27-s + 0.657·37-s + 0.640·39-s + 1.16·47-s + 6/7·49-s + 1.04·59-s + 0.512·61-s − 0.963·69-s + 0.949·71-s − 0.468·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s − 0.773·107-s − 1.91·109-s − 0.379·111-s − 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300683416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300683416\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 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
−8.620648964698798042027995861180, −8.069296003024719182286593042392, −7.59973622177298338712485066709, −7.11596934108936843626981527026, −6.85707424804806269299182215242, −6.33553372404088741939001164645, −5.61573405155964112907825761587, −5.32453342739906912265969317730, −4.93163605770324517672365246049, −4.14022431258350108091479787197, −3.94743574841707661451444935200, −2.81721097613211638420411402400, −2.62083869433478434502766859076, −1.59209964826416918235844898746, −0.65618734072754594901705452172,
0.65618734072754594901705452172, 1.59209964826416918235844898746, 2.62083869433478434502766859076, 2.81721097613211638420411402400, 3.94743574841707661451444935200, 4.14022431258350108091479787197, 4.93163605770324517672365246049, 5.32453342739906912265969317730, 5.61573405155964112907825761587, 6.33553372404088741939001164645, 6.85707424804806269299182215242, 7.11596934108936843626981527026, 7.59973622177298338712485066709, 8.069296003024719182286593042392, 8.620648964698798042027995861180
