| L(s) = 1 | + 2·2-s + 3-s − 4-s + 2·5-s + 2·6-s − 2·7-s − 8·8-s − 4·9-s + 4·10-s − 12-s + 7·13-s − 4·14-s + 2·15-s − 7·16-s − 7·17-s − 8·18-s + 6·19-s − 2·20-s − 2·21-s − 8·24-s + 3·25-s + 14·26-s − 6·27-s + 2·28-s − 3·29-s + 4·30-s − 2·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s − 2.82·8-s − 4/3·9-s + 1.26·10-s − 0.288·12-s + 1.94·13-s − 1.06·14-s + 0.516·15-s − 7/4·16-s − 1.69·17-s − 1.88·18-s + 1.37·19-s − 0.447·20-s − 0.436·21-s − 1.63·24-s + 3/5·25-s + 2.74·26-s − 1.15·27-s + 0.377·28-s − 0.557·29-s + 0.730·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 T + 186 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 166 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 41 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 177 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 213 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−8.363869097061957795407803501729, −8.090635078334848634987456926726, −7.41730184945329260695762589699, −6.99273130110809780772425798926, −6.27057720483562759709937403814, −6.19593417256415516430699187716, −5.91968739664113466366855752065, −5.82676844418015295205330549423, −5.01384306356432047680339186342, −4.99246631350684430683019928171, −4.43558755365572538895601150975, −4.02242529830773172811397436328, −3.46975542500956125596302321443, −3.27094973841425001881288726737, −2.90666106317496093209295830419, −2.72678373807638414098944539578, −1.72588541897764349756496495836, −1.33775117328207920081275777630, 0, 0,
1.33775117328207920081275777630, 1.72588541897764349756496495836, 2.72678373807638414098944539578, 2.90666106317496093209295830419, 3.27094973841425001881288726737, 3.46975542500956125596302321443, 4.02242529830773172811397436328, 4.43558755365572538895601150975, 4.99246631350684430683019928171, 5.01384306356432047680339186342, 5.82676844418015295205330549423, 5.91968739664113466366855752065, 6.19593417256415516430699187716, 6.27057720483562759709937403814, 6.99273130110809780772425798926, 7.41730184945329260695762589699, 8.090635078334848634987456926726, 8.363869097061957795407803501729
