| L(s) = 1 | + 3-s − 6·5-s + 9-s − 6·15-s − 6·17-s + 2·19-s + 17·25-s + 27-s + 10·31-s − 6·45-s − 49-s − 6·51-s + 2·57-s − 6·59-s − 14·61-s + 16·67-s + 18·71-s − 2·73-s + 17·75-s − 8·79-s + 81-s + 36·85-s + 10·93-s − 12·95-s + 4·103-s + 24·107-s − 121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.68·5-s + 1/3·9-s − 1.54·15-s − 1.45·17-s + 0.458·19-s + 17/5·25-s + 0.192·27-s + 1.79·31-s − 0.894·45-s − 1/7·49-s − 0.840·51-s + 0.264·57-s − 0.781·59-s − 1.79·61-s + 1.95·67-s + 2.13·71-s − 0.234·73-s + 1.96·75-s − 0.900·79-s + 1/9·81-s + 3.90·85-s + 1.03·93-s − 1.23·95-s + 0.394·103-s + 2.32·107-s − 0.0909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.260950774398169693455935936530, −7.73176510304631626350186455297, −7.41609012590777110143141777255, −7.03313481593270888213314588827, −6.47388089705755043923762089426, −6.09169394833089910229281190325, −5.03899104096684628908870883654, −4.69502562339021796938954572316, −4.30079091902536361808475523480, −3.81836826184895820813585011158, −3.41372916558680725837243675660, −2.85471881407479794663936058223, −2.15302361248508074428644861260, −0.951161956955515466584108883706, 0,
0.951161956955515466584108883706, 2.15302361248508074428644861260, 2.85471881407479794663936058223, 3.41372916558680725837243675660, 3.81836826184895820813585011158, 4.30079091902536361808475523480, 4.69502562339021796938954572316, 5.03899104096684628908870883654, 6.09169394833089910229281190325, 6.47388089705755043923762089426, 7.03313481593270888213314588827, 7.41609012590777110143141777255, 7.73176510304631626350186455297, 8.260950774398169693455935936530
