| L(s) = 1 | + 9-s + 6·13-s + 4·17-s − 6·25-s + 4·29-s + 2·49-s + 4·53-s − 4·61-s − 24·79-s + 81-s + 4·101-s + 8·103-s + 16·107-s + 20·113-s + 6·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 23·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.66·13-s + 0.970·17-s − 6/5·25-s + 0.742·29-s + 2/7·49-s + 0.549·53-s − 0.512·61-s − 2.70·79-s + 1/9·81-s + 0.398·101-s + 0.788·103-s + 1.54·107-s + 1.88·113-s + 0.554·117-s + 0.909·121-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 + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719115326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719115326\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−9.699438769496114402033257178004, −8.993860452341910821794806769078, −8.589788743802462385769043811054, −8.215799634359465414661645965473, −7.53440663921409189155491314773, −7.22581679137061663285206796479, −6.43986804523822436911164629458, −5.94124585537876212909233098247, −5.67153147242154355347872008580, −4.80024266791599446847212426264, −4.18301772950860384666382204939, −3.59866243929372867606894540722, −3.03644367581316318535608327538, −1.94677222991202967287945155352, −1.07453461325392778034782218303,
1.07453461325392778034782218303, 1.94677222991202967287945155352, 3.03644367581316318535608327538, 3.59866243929372867606894540722, 4.18301772950860384666382204939, 4.80024266791599446847212426264, 5.67153147242154355347872008580, 5.94124585537876212909233098247, 6.43986804523822436911164629458, 7.22581679137061663285206796479, 7.53440663921409189155491314773, 8.215799634359465414661645965473, 8.589788743802462385769043811054, 8.993860452341910821794806769078, 9.699438769496114402033257178004
