| L(s) = 1 | + 4·3-s + 8·9-s − 4·11-s − 12·17-s + 12·19-s + 12·27-s − 16·33-s + 12·43-s + 14·49-s − 48·51-s + 48·57-s + 20·59-s − 12·67-s + 23·81-s − 4·83-s − 20·97-s − 32·99-s + 28·107-s − 36·113-s + 8·121-s + 127-s + 48·129-s + 131-s + 137-s + 139-s + 56·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 1.20·11-s − 2.91·17-s + 2.75·19-s + 2.30·27-s − 2.78·33-s + 1.82·43-s + 2·49-s − 6.72·51-s + 6.35·57-s + 2.60·59-s − 1.46·67-s + 23/9·81-s − 0.439·83-s − 2.03·97-s − 3.21·99-s + 2.70·107-s − 3.38·113-s + 8/11·121-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.61·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.486587907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.486587907\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.00167753879394501391472893078, −9.434506558798499245890033369067, −9.284251650252961596063019202131, −8.803849558941571265239170047156, −8.698799544335503279115229005473, −8.020284198885914417639769930380, −7.88182023823245553295176695194, −7.23827562527942798807709696167, −7.11827242659521147135499863243, −6.62717317503173937190731280993, −5.67865443344435722311545911789, −5.47715606407333095782087229301, −4.77111021245306703550164216763, −4.20231806608114379710959866625, −3.92442608162171711703254581100, −3.05371656727296707824483995781, −2.88459125440640351331937796670, −2.36445753953027619438674644962, −1.98773776291656032079881168238, −0.834582311387276827991493732713,
0.834582311387276827991493732713, 1.98773776291656032079881168238, 2.36445753953027619438674644962, 2.88459125440640351331937796670, 3.05371656727296707824483995781, 3.92442608162171711703254581100, 4.20231806608114379710959866625, 4.77111021245306703550164216763, 5.47715606407333095782087229301, 5.67865443344435722311545911789, 6.62717317503173937190731280993, 7.11827242659521147135499863243, 7.23827562527942798807709696167, 7.88182023823245553295176695194, 8.020284198885914417639769930380, 8.698799544335503279115229005473, 8.803849558941571265239170047156, 9.284251650252961596063019202131, 9.434506558798499245890033369067, 10.00167753879394501391472893078
