| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 8·15-s + 4·17-s − 8·19-s + 2·23-s + 4·25-s + 4·27-s + 12·29-s − 8·31-s − 12·37-s − 4·41-s − 8·43-s + 12·45-s − 4·47-s − 12·49-s + 8·51-s + 12·53-s − 16·57-s − 12·59-s + 12·61-s − 24·67-s + 4·69-s + 12·73-s + 8·75-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 2.06·15-s + 0.970·17-s − 1.83·19-s + 0.417·23-s + 4/5·25-s + 0.769·27-s + 2.22·29-s − 1.43·31-s − 1.97·37-s − 0.624·41-s − 1.21·43-s + 1.78·45-s − 0.583·47-s − 1.71·49-s + 1.12·51-s + 1.64·53-s − 2.11·57-s − 1.56·59-s + 1.53·61-s − 2.93·67-s + 0.481·69-s + 1.40·73-s + 0.923·75-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.927953941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.927953941\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 260 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 164 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−12.12044911188866483137545613273, −11.90050735963500801445277864692, −10.90679843816514506236089674569, −10.43719971511234173145824422447, −10.13254974027246679309843872052, −9.882319819910005822888162400279, −9.158602269229153462340723257850, −8.872006503893159585757153755981, −8.384701073411926562626182233033, −7.970652796696304863211628472025, −7.16598235489499128142264135198, −6.60296333474831789043564547276, −6.31670944000890475573566842943, −5.52966722596093998967060465943, −5.04545212006696388313034472462, −4.35367144990027765798827833257, −3.43589858415556779874635487995, −2.94301323238943509564118516416, −1.95476673853503459838914818979, −1.70534628834130459930649122147,
1.70534628834130459930649122147, 1.95476673853503459838914818979, 2.94301323238943509564118516416, 3.43589858415556779874635487995, 4.35367144990027765798827833257, 5.04545212006696388313034472462, 5.52966722596093998967060465943, 6.31670944000890475573566842943, 6.60296333474831789043564547276, 7.16598235489499128142264135198, 7.970652796696304863211628472025, 8.384701073411926562626182233033, 8.872006503893159585757153755981, 9.158602269229153462340723257850, 9.882319819910005822888162400279, 10.13254974027246679309843872052, 10.43719971511234173145824422447, 10.90679843816514506236089674569, 11.90050735963500801445277864692, 12.12044911188866483137545613273
