| L(s) = 1 | − 4-s − 5-s − 3·9-s + 7·11-s − 3·16-s + 8·19-s + 20-s + 25-s − 3·29-s + 2·31-s + 3·36-s − 41-s − 7·44-s + 3·45-s + 49-s − 7·55-s + 5·59-s + 21·61-s + 7·64-s + 24·71-s − 8·76-s − 10·79-s + 3·80-s + 2·89-s − 8·95-s − 21·99-s − 100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.447·5-s − 9-s + 2.11·11-s − 3/4·16-s + 1.83·19-s + 0.223·20-s + 1/5·25-s − 0.557·29-s + 0.359·31-s + 1/2·36-s − 0.156·41-s − 1.05·44-s + 0.447·45-s + 1/7·49-s − 0.943·55-s + 0.650·59-s + 2.68·61-s + 7/8·64-s + 2.84·71-s − 0.917·76-s − 1.12·79-s + 0.335·80-s + 0.211·89-s − 0.820·95-s − 2.11·99-s − 0.0999·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106456727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106456727\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 331 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
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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
−10.01499244692265779382834373670, −9.597287612587930727221694461997, −9.274392963078847225722112804596, −8.649679793638444255697139727111, −8.410384465840586370523201176541, −7.63232589238166029348888745498, −6.93310706619865061972954794948, −6.66540022484453257831025472625, −5.82930486592205458844674091138, −5.29122317955955096690201529733, −4.61557275276563948105644133743, −3.76313549857495181722161915381, −3.52425382628233441606004274867, −2.38438337170026039176009959753, −1.01061228574463095034521581981,
1.01061228574463095034521581981, 2.38438337170026039176009959753, 3.52425382628233441606004274867, 3.76313549857495181722161915381, 4.61557275276563948105644133743, 5.29122317955955096690201529733, 5.82930486592205458844674091138, 6.66540022484453257831025472625, 6.93310706619865061972954794948, 7.63232589238166029348888745498, 8.410384465840586370523201176541, 8.649679793638444255697139727111, 9.274392963078847225722112804596, 9.597287612587930727221694461997, 10.01499244692265779382834373670
