| L(s) = 1 | + 4·5-s + 4·17-s − 2·19-s + 2·25-s + 12·31-s + 6·49-s + 16·59-s − 16·61-s − 4·67-s + 16·71-s − 16·79-s + 16·85-s − 8·95-s − 12·101-s − 16·103-s − 40·107-s + 20·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.970·17-s − 0.458·19-s + 2/5·25-s + 2.15·31-s + 6/7·49-s + 2.08·59-s − 2.04·61-s − 0.488·67-s + 1.89·71-s − 1.80·79-s + 1.73·85-s − 0.820·95-s − 1.19·101-s − 1.57·103-s − 3.86·107-s + 1.81·121-s − 2.50·125-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 + 3.85·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.998179179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.998179179\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 186 T^{2} + p^{2} T^{4} \) |
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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
−9.113230658535619010469194486066, −8.628418519873205653505689124771, −8.247950402571882068434279879202, −7.979024601144416548478720392696, −7.57928460845954272202522472185, −6.93046491031248673035674699290, −6.54422947909739461180924384449, −6.51730782226577339672886193519, −5.62804488175347957839789838602, −5.62177203256154037941756271806, −5.53216175885071598134879869818, −4.68256906960174965616361121885, −4.30052410883299608639780508916, −3.96150018981954052938822647874, −3.13231266049494287666858825674, −2.85744984246124962714683867956, −2.32802530422637219936668147700, −1.83543257837750060927044170708, −1.36328994955260503026483027157, −0.66004913437722848509469054681,
0.66004913437722848509469054681, 1.36328994955260503026483027157, 1.83543257837750060927044170708, 2.32802530422637219936668147700, 2.85744984246124962714683867956, 3.13231266049494287666858825674, 3.96150018981954052938822647874, 4.30052410883299608639780508916, 4.68256906960174965616361121885, 5.53216175885071598134879869818, 5.62177203256154037941756271806, 5.62804488175347957839789838602, 6.51730782226577339672886193519, 6.54422947909739461180924384449, 6.93046491031248673035674699290, 7.57928460845954272202522472185, 7.979024601144416548478720392696, 8.247950402571882068434279879202, 8.628418519873205653505689124771, 9.113230658535619010469194486066
