| L(s) = 1 | − 4·3-s − 8·5-s + 8·9-s + 4·11-s + 32·15-s + 7·16-s − 4·23-s + 38·25-s − 20·27-s + 8·31-s − 16·33-s + 12·37-s − 64·45-s + 12·47-s − 28·48-s − 4·53-s − 32·55-s + 12·67-s + 16·69-s − 32·71-s − 152·75-s − 56·80-s + 50·81-s − 32·93-s − 28·97-s + 32·99-s + 36·103-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.57·5-s + 8/3·9-s + 1.20·11-s + 8.26·15-s + 7/4·16-s − 0.834·23-s + 38/5·25-s − 3.84·27-s + 1.43·31-s − 2.78·33-s + 1.97·37-s − 9.54·45-s + 1.75·47-s − 4.04·48-s − 0.549·53-s − 4.31·55-s + 1.46·67-s + 1.92·69-s − 3.79·71-s − 17.5·75-s − 6.26·80-s + 50/9·81-s − 3.31·93-s − 2.84·97-s + 3.21·99-s + 3.54·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1497366223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1497366223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^3$ | \( 1 - 382 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 5218 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 6382 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.52091864690022716191318947871, −11.32660465527116186695832321215, −11.08179773469017204230085075962, −10.89944174647849326019021614739, −10.43076605168957514655098384351, −9.993905212072615523114276494335, −9.898819583401376212631158583627, −9.321113994849563190296681180799, −8.743308677963307139727859113879, −8.555156740161672823066773541136, −8.062227137583947885893163577620, −7.84273566115687712373307112076, −7.41604196646297359714148104156, −7.38974029877214882236176071903, −6.93234000252375324452646692331, −6.27960050060641700670605386254, −6.13754429632765401532898454305, −5.72639588141454045701669712650, −5.30018529982774542597949289403, −4.45905546606990116365670907887, −4.43129273790414915534528716769, −4.11659020628383720329043507720, −3.56769015484041802071715761194, −3.13099997286512682748889805950, −0.872296453380042686988010080295,
0.872296453380042686988010080295, 3.13099997286512682748889805950, 3.56769015484041802071715761194, 4.11659020628383720329043507720, 4.43129273790414915534528716769, 4.45905546606990116365670907887, 5.30018529982774542597949289403, 5.72639588141454045701669712650, 6.13754429632765401532898454305, 6.27960050060641700670605386254, 6.93234000252375324452646692331, 7.38974029877214882236176071903, 7.41604196646297359714148104156, 7.84273566115687712373307112076, 8.062227137583947885893163577620, 8.555156740161672823066773541136, 8.743308677963307139727859113879, 9.321113994849563190296681180799, 9.898819583401376212631158583627, 9.993905212072615523114276494335, 10.43076605168957514655098384351, 10.89944174647849326019021614739, 11.08179773469017204230085075962, 11.32660465527116186695832321215, 11.52091864690022716191318947871
