| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 7-s − 5·9-s − 2·12-s − 4·13-s − 2·14-s − 4·16-s + 8·17-s + 10·18-s − 21-s + 2·25-s + 8·26-s + 8·27-s + 2·28-s − 5·31-s + 8·32-s − 16·34-s − 10·36-s + 4·39-s + 2·42-s + 4·48-s − 6·49-s − 4·50-s − 8·51-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s − 5/3·9-s − 0.577·12-s − 1.10·13-s − 0.534·14-s − 16-s + 1.94·17-s + 2.35·18-s − 0.218·21-s + 2/5·25-s + 1.56·26-s + 1.53·27-s + 0.377·28-s − 0.898·31-s + 1.41·32-s − 2.74·34-s − 5/3·36-s + 0.640·39-s + 0.308·42-s + 0.577·48-s − 6/7·49-s − 0.565·50-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 31 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.542261679545153034511562971660, −8.027484448136311172438005590303, −7.63660857316058925171011936563, −7.37259907961845724310157448571, −6.65325877714715521196587482512, −6.20965674234546691466533925303, −5.60085696382095652293964293735, −5.14717296640011951291422714465, −4.92490369052887204837084680332, −3.96024968567489685238146643741, −3.16070124367081256827512751772, −2.69168539466595962652570841749, −1.88104253334614448379493415708, −0.951078809257007125955417953695, 0,
0.951078809257007125955417953695, 1.88104253334614448379493415708, 2.69168539466595962652570841749, 3.16070124367081256827512751772, 3.96024968567489685238146643741, 4.92490369052887204837084680332, 5.14717296640011951291422714465, 5.60085696382095652293964293735, 6.20965674234546691466533925303, 6.65325877714715521196587482512, 7.37259907961845724310157448571, 7.63660857316058925171011936563, 8.027484448136311172438005590303, 8.542261679545153034511562971660
