| L(s) = 1 | − 20·7-s − 9·9-s + 28·17-s − 296·23-s + 234·25-s + 36·31-s + 756·41-s + 296·47-s − 386·49-s + 180·63-s − 1.08e3·71-s + 2.03e3·73-s + 772·79-s + 81·81-s + 764·89-s + 596·97-s + 1.63e3·103-s + 3.71e3·113-s − 560·119-s + 2.64e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 252·153-s + ⋯ |
| L(s) = 1 | − 1.07·7-s − 1/3·9-s + 0.399·17-s − 2.68·23-s + 1.87·25-s + 0.208·31-s + 2.87·41-s + 0.918·47-s − 1.12·49-s + 0.359·63-s − 1.80·71-s + 3.26·73-s + 1.09·79-s + 1/9·81-s + 0.909·89-s + 0.623·97-s + 1.56·103-s + 3.09·113-s − 0.431·119-s + 1.98·121-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 − 0.133·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.943813198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943813198\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )( 1 + 22 T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2646 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 p^{2} T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13654 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 148 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 43594 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32662 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 27610 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 148 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 168154 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 227574 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 258598 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 122662 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 540 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1018 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1131910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 382 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 298 T + p^{3} T^{2} )^{2} \) |
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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
−10.06412880286996104546043601243, −9.751388033233978322473583321122, −9.229882828112835988448382519621, −9.087367853985456437630971938104, −8.289995744138593041751743642622, −8.123863695769537343772062747455, −7.51580787808757577101812633957, −7.19581808239193882201188116228, −6.44348574757048780674800796454, −6.21015886587170321354814721995, −5.90290894069193182591688184725, −5.31366373908271420382427563834, −4.59438080431728914233571929504, −4.28319418240562268281295433380, −3.42411773794962452456052146627, −3.33562161633800997484863212547, −2.42237892918253509702354656399, −2.10881593762887451814678537571, −0.975360267827377418771369908872, −0.45335486353941306838209751755,
0.45335486353941306838209751755, 0.975360267827377418771369908872, 2.10881593762887451814678537571, 2.42237892918253509702354656399, 3.33562161633800997484863212547, 3.42411773794962452456052146627, 4.28319418240562268281295433380, 4.59438080431728914233571929504, 5.31366373908271420382427563834, 5.90290894069193182591688184725, 6.21015886587170321354814721995, 6.44348574757048780674800796454, 7.19581808239193882201188116228, 7.51580787808757577101812633957, 8.123863695769537343772062747455, 8.289995744138593041751743642622, 9.087367853985456437630971938104, 9.229882828112835988448382519621, 9.751388033233978322473583321122, 10.06412880286996104546043601243
