| L(s) = 1 | + 54·9-s − 188·17-s − 234·25-s + 460·41-s − 686·49-s − 2.19e3·73-s + 2.18e3·81-s + 3.34e3·89-s + 1.18e3·97-s + 4.00e3·113-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.01e4·153-s + 157-s + 163-s + 167-s + 4.07e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·9-s − 2.68·17-s − 1.87·25-s + 1.75·41-s − 2·49-s − 3.52·73-s + 3·81-s + 3.97·89-s + 1.24·97-s + 3.33·113-s + 2·121-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 − 5.36·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.85·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.057944192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057944192\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )( 1 + 4 T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )( 1 + 92 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 94 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )( 1 + 284 T + p^{3} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 572 T + p^{3} T^{2} )( 1 + 572 T + p^{3} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 468 T + p^{3} T^{2} )( 1 + 468 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1098 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1670 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 594 T + p^{3} T^{2} )^{2} \) |
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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
−11.66901922938060103720676978148, −11.37481926749180207954798453231, −10.88255687756129246859806935396, −10.24073929299626151905615149063, −10.05164780331912281733296421340, −9.237975691032134259323431376684, −9.201052795409560342752419849890, −8.438948090397959762662563354947, −7.71153463799102336656097111050, −7.41303768092377957806265172261, −6.88299932706972729417178247930, −6.27537257385577233525892743991, −5.95877907449412782513055676073, −4.79557237087065021850700775283, −4.47426015712326044310432944421, −4.11581531625886349356492147636, −3.26612440899367609296281861267, −2.02717249790339924661933262425, −1.87492467183010786537638458261, −0.55858118660512964879006640911,
0.55858118660512964879006640911, 1.87492467183010786537638458261, 2.02717249790339924661933262425, 3.26612440899367609296281861267, 4.11581531625886349356492147636, 4.47426015712326044310432944421, 4.79557237087065021850700775283, 5.95877907449412782513055676073, 6.27537257385577233525892743991, 6.88299932706972729417178247930, 7.41303768092377957806265172261, 7.71153463799102336656097111050, 8.438948090397959762662563354947, 9.201052795409560342752419849890, 9.237975691032134259323431376684, 10.05164780331912281733296421340, 10.24073929299626151905615149063, 10.88255687756129246859806935396, 11.37481926749180207954798453231, 11.66901922938060103720676978148
