| L(s) = 1 | − 46·9-s + 32·11-s − 248·19-s − 284·29-s + 376·31-s + 108·41-s + 362·49-s + 1.12e3·59-s − 524·61-s − 280·71-s − 2.32e3·79-s + 1.38e3·81-s + 1.70e3·89-s − 1.47e3·99-s − 3.58e3·101-s − 1.33e3·109-s − 1.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + ⋯ |
| L(s) = 1 | − 1.70·9-s + 0.877·11-s − 2.99·19-s − 1.81·29-s + 2.17·31-s + 0.411·41-s + 1.05·49-s + 2.48·59-s − 1.09·61-s − 0.468·71-s − 3.30·79-s + 1.90·81-s + 2.03·89-s − 1.49·99-s − 3.53·101-s − 1.17·109-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 + 1.98·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7262057353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7262057353\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 46 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 362 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9790 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22570 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 142 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 188 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 60502 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 154658 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 206202 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 246890 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 564 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 294610 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 731410 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 854 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1596862 T^{2} + p^{6} T^{4} \) |
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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
−11.36127089573449341353168937740, −10.50596259776510677272078276146, −10.42113049988347821360902109371, −9.670142889132792129962842750992, −8.995370308465347412674345689470, −8.897801567631986704144003511665, −8.288440232768443102967695216345, −8.141225887472297892316593813873, −7.32012793960520685771887357082, −6.62772072484578481804569170687, −6.40763493203410626874565291398, −5.80103958506274685395219088066, −5.49548108676700692797844708316, −4.57180729496062717471175468585, −4.12649830905171538408970496194, −3.65056409752179955135861437828, −2.58619272534543612444222421437, −2.44900526284623493229656373819, −1.41134842858219055373414203222, −0.28424045383952980273618341163,
0.28424045383952980273618341163, 1.41134842858219055373414203222, 2.44900526284623493229656373819, 2.58619272534543612444222421437, 3.65056409752179955135861437828, 4.12649830905171538408970496194, 4.57180729496062717471175468585, 5.49548108676700692797844708316, 5.80103958506274685395219088066, 6.40763493203410626874565291398, 6.62772072484578481804569170687, 7.32012793960520685771887357082, 8.141225887472297892316593813873, 8.288440232768443102967695216345, 8.897801567631986704144003511665, 8.995370308465347412674345689470, 9.670142889132792129962842750992, 10.42113049988347821360902109371, 10.50596259776510677272078276146, 11.36127089573449341353168937740
