| L(s) = 1 | − 32·11-s − 8·19-s + 164·29-s − 16·31-s + 492·41-s + 286·49-s − 1.18e3·59-s + 1.14e3·61-s − 1.53e3·71-s − 816·79-s − 1.02e3·89-s − 1.33e3·101-s − 4.15e3·109-s − 1.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.877·11-s − 0.0965·19-s + 1.05·29-s − 0.0926·31-s + 1.87·41-s + 0.833·49-s − 2.61·59-s + 2.40·61-s − 2.56·71-s − 1.16·79-s − 1.21·89-s − 1.31·101-s − 3.65·109-s − 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.468·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3898744643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3898744643\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 80170 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 115562 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 195050 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 592 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 574 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 571942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 466670 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1116678 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 510 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1561150 T^{2} + p^{6} T^{4} \) |
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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
−9.288114226828239293753977262184, −8.692750745535502815404172927081, −8.253321698021315909811104311238, −7.996808830802879189537972607747, −7.53046943296267950319079240310, −7.20899206776394810021968906343, −6.77886743514006506286687694441, −6.22347300470339390118252339297, −5.98064069019703983946956183829, −5.32678184117386842834354431399, −5.24497244656758590966007863730, −4.48329925214013270137423184135, −4.22040070059689329071241077562, −3.77859865402879561998910321737, −3.00211465037246109890376804432, −2.62864162346523547279153904977, −2.40845661680501109674956158571, −1.36665556468595375922304440596, −1.16557878145854026123175189443, −0.13793380440452283220858643500,
0.13793380440452283220858643500, 1.16557878145854026123175189443, 1.36665556468595375922304440596, 2.40845661680501109674956158571, 2.62864162346523547279153904977, 3.00211465037246109890376804432, 3.77859865402879561998910321737, 4.22040070059689329071241077562, 4.48329925214013270137423184135, 5.24497244656758590966007863730, 5.32678184117386842834354431399, 5.98064069019703983946956183829, 6.22347300470339390118252339297, 6.77886743514006506286687694441, 7.20899206776394810021968906343, 7.53046943296267950319079240310, 7.996808830802879189537972607747, 8.253321698021315909811104311238, 8.692750745535502815404172927081, 9.288114226828239293753977262184
