| L(s) = 1 | + 2·7-s − 3·9-s − 10·17-s − 6·23-s − 6·25-s + 20·31-s − 12·41-s + 16·47-s − 11·49-s − 6·63-s − 24·71-s + 22·73-s + 32·79-s − 8·89-s − 24·97-s − 4·103-s + 16·113-s − 20·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 2.42·17-s − 1.25·23-s − 6/5·25-s + 3.59·31-s − 1.87·41-s + 2.33·47-s − 1.57·49-s − 0.755·63-s − 2.84·71-s + 2.57·73-s + 3.60·79-s − 0.847·89-s − 2.43·97-s − 0.394·103-s + 1.50·113-s − 1.83·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213666843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213666843\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−10.34162321963084756228495507212, −9.789944851281245100404629534036, −8.888684763342553795967179645308, −8.702666780100205789920154436905, −8.349699421063129414052767235301, −7.931113381001288443833542932298, −7.74981057464855372059181133882, −6.82861611084092418158079627366, −6.66782394428390095259465866418, −6.09620023331127049825297063168, −5.97086886665497760988066145289, −5.06095169685955457986079072353, −4.92058352265273508167847449905, −4.23343298498192265577536023043, −4.07736104185211590370685136133, −3.23270879701863519368115653074, −2.55192185855878262092093367963, −2.25676566650142583540662821214, −1.59768397922571576498652599915, −0.45615319848583992520295124011,
0.45615319848583992520295124011, 1.59768397922571576498652599915, 2.25676566650142583540662821214, 2.55192185855878262092093367963, 3.23270879701863519368115653074, 4.07736104185211590370685136133, 4.23343298498192265577536023043, 4.92058352265273508167847449905, 5.06095169685955457986079072353, 5.97086886665497760988066145289, 6.09620023331127049825297063168, 6.66782394428390095259465866418, 6.82861611084092418158079627366, 7.74981057464855372059181133882, 7.931113381001288443833542932298, 8.349699421063129414052767235301, 8.702666780100205789920154436905, 8.888684763342553795967179645308, 9.789944851281245100404629534036, 10.34162321963084756228495507212
