| L(s) = 1 | − 3·9-s − 4·13-s + 6·17-s + 4·23-s + 8·25-s − 2·29-s + 12·43-s − 2·49-s + 2·53-s − 8·61-s + 8·79-s + 9·81-s − 6·101-s − 4·103-s − 4·107-s + 10·113-s + 12·117-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 18·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 9-s − 1.10·13-s + 1.45·17-s + 0.834·23-s + 8/5·25-s − 0.371·29-s + 1.82·43-s − 2/7·49-s + 0.274·53-s − 1.02·61-s + 0.900·79-s + 81-s − 0.597·101-s − 0.394·103-s − 0.386·107-s + 0.940·113-s + 1.10·117-s + 1.09·121-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 − 1.45·153-s + 0.0798·157-s + 0.0783·163-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
| \(L(1)\) |
\(\approx\) |
\(1.359310864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359310864\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
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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
−9.526004404431178876031710043647, −9.164352602874733233299723311120, −8.619911126044199085540764788205, −8.125918215506294645071474286457, −7.55388770150314928754095984897, −7.18330410569255458878422151340, −6.61942314254216046873127828148, −5.80803383582238864941480515497, −5.54991894915586879814361046912, −4.88664124590497733589586684613, −4.39987646534527496436176329610, −3.30614312674463841973957421169, −3.02121918077198931080455712253, −2.20439711277883076301974744262, −0.882561622700950431239185887106,
0.882561622700950431239185887106, 2.20439711277883076301974744262, 3.02121918077198931080455712253, 3.30614312674463841973957421169, 4.39987646534527496436176329610, 4.88664124590497733589586684613, 5.54991894915586879814361046912, 5.80803383582238864941480515497, 6.61942314254216046873127828148, 7.18330410569255458878422151340, 7.55388770150314928754095984897, 8.125918215506294645071474286457, 8.619911126044199085540764788205, 9.164352602874733233299723311120, 9.526004404431178876031710043647
