| L(s) = 1 | + 50·25-s − 94·37-s − 166·43-s − 218·67-s + 262·79-s − 286·109-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 337·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 2·25-s − 2.54·37-s − 3.86·43-s − 3.25·67-s + 3.31·79-s − 2.62·109-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.99·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8653182710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8653182710\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 59 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 83 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 74 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 109 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 + 143 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−9.167619524977627984786768467535, −8.893507568642603225504536023398, −8.618561865501307005566447917880, −8.138245616333137025369813177963, −7.82371266056352850905535584450, −7.26591815117370438036045134988, −6.84246275220602777596355889579, −6.49502437640926880484117657050, −6.40687313990306286913915338816, −5.44770955865084735282240718119, −5.23686254162897457698565427796, −4.88478739772536426627467866995, −4.49605238053017678983952142430, −3.60160788491019171088966842766, −3.53004825071167767830153317139, −2.92255987812982272530874233845, −2.40210214058272971773926815179, −1.51759060196703476486870935817, −1.39437209184315243454134998295, −0.24362651316094292010829520575,
0.24362651316094292010829520575, 1.39437209184315243454134998295, 1.51759060196703476486870935817, 2.40210214058272971773926815179, 2.92255987812982272530874233845, 3.53004825071167767830153317139, 3.60160788491019171088966842766, 4.49605238053017678983952142430, 4.88478739772536426627467866995, 5.23686254162897457698565427796, 5.44770955865084735282240718119, 6.40687313990306286913915338816, 6.49502437640926880484117657050, 6.84246275220602777596355889579, 7.26591815117370438036045134988, 7.82371266056352850905535584450, 8.138245616333137025369813177963, 8.618561865501307005566447917880, 8.893507568642603225504536023398, 9.167619524977627984786768467535
