| L(s) = 1 | − 4-s − 6·11-s + 16-s − 4·19-s − 12·29-s − 20·31-s − 12·41-s + 6·44-s − 2·49-s + 6·59-s − 14·61-s − 64-s + 30·71-s + 4·76-s − 16·79-s − 24·89-s + 24·101-s − 4·109-s + 12·116-s + 5·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.80·11-s + 1/4·16-s − 0.917·19-s − 2.22·29-s − 3.59·31-s − 1.87·41-s + 0.904·44-s − 2/7·49-s + 0.781·59-s − 1.79·61-s − 1/8·64-s + 3.56·71-s + 0.458·76-s − 1.80·79-s − 2.54·89-s + 2.38·101-s − 0.383·109-s + 1.11·116-s + 5/11·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−9.358767197798941468871957599282, −8.969807269776292482679102854866, −8.678601480541626286136654443722, −8.187238228461980480405173279197, −7.75135484508714870051209006322, −7.44610568031542127505266294045, −7.11732366715655303509778126209, −6.55914519094423828045155233596, −5.93172244312131361664149722274, −5.52439635985594017990594264392, −5.14095944070439049398360999604, −5.04250681498774848020633839555, −4.12813381427571540358976871030, −3.74921696363360829586538440219, −3.39469069070105949874617725048, −2.64802911163887194713286263259, −2.00805247241882622671154375703, −1.64119946306213127505685108561, 0, 0,
1.64119946306213127505685108561, 2.00805247241882622671154375703, 2.64802911163887194713286263259, 3.39469069070105949874617725048, 3.74921696363360829586538440219, 4.12813381427571540358976871030, 5.04250681498774848020633839555, 5.14095944070439049398360999604, 5.52439635985594017990594264392, 5.93172244312131361664149722274, 6.55914519094423828045155233596, 7.11732366715655303509778126209, 7.44610568031542127505266294045, 7.75135484508714870051209006322, 8.187238228461980480405173279197, 8.678601480541626286136654443722, 8.969807269776292482679102854866, 9.358767197798941468871957599282
