| L(s) = 1 | + 3-s + 9-s + 11-s + 8·17-s − 6·25-s + 27-s − 16·29-s − 16·31-s + 33-s + 20·37-s + 16·41-s − 10·49-s + 8·51-s + 24·67-s − 6·75-s + 81-s + 32·83-s − 16·87-s − 16·93-s − 4·97-s + 99-s − 32·101-s + 8·103-s + 20·111-s + 121-s + 16·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.94·17-s − 6/5·25-s + 0.192·27-s − 2.97·29-s − 2.87·31-s + 0.174·33-s + 3.28·37-s + 2.49·41-s − 1.42·49-s + 1.12·51-s + 2.93·67-s − 0.692·75-s + 1/9·81-s + 3.51·83-s − 1.71·87-s − 1.65·93-s − 0.406·97-s + 0.100·99-s − 3.18·101-s + 0.788·103-s + 1.89·111-s + 1/11·121-s + 1.44·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.456613332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456613332\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.074379069264153068708952280908, −7.996057650348538062805293058353, −7.66700790333795020456742124482, −7.30223796795303190064148405091, −6.70453643193197308588262041684, −5.95818902421944129428630267000, −5.57273521417705379772496968279, −5.51331429337006714198744949212, −4.52702091907548133307727064364, −3.79955930348242148717135072932, −3.79294235641559942977546839028, −3.11474424285935633940176598416, −2.21252199435586523776406167130, −1.82500088072096369066062448326, −0.796015146435045811880856861272,
0.796015146435045811880856861272, 1.82500088072096369066062448326, 2.21252199435586523776406167130, 3.11474424285935633940176598416, 3.79294235641559942977546839028, 3.79955930348242148717135072932, 4.52702091907548133307727064364, 5.51331429337006714198744949212, 5.57273521417705379772496968279, 5.95818902421944129428630267000, 6.70453643193197308588262041684, 7.30223796795303190064148405091, 7.66700790333795020456742124482, 7.996057650348538062805293058353, 8.074379069264153068708952280908
