| L(s) = 1 | − 2·5-s − 4·7-s − 4·11-s − 4·13-s + 8·17-s + 4·19-s − 4·23-s + 3·25-s + 10·29-s + 4·31-s + 8·35-s + 6·41-s + 16·43-s − 4·47-s + 49-s + 12·53-s + 8·55-s − 8·59-s − 10·61-s + 8·65-s − 16·67-s − 12·71-s + 16·77-s − 24·79-s + 16·83-s − 16·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 1.20·11-s − 1.10·13-s + 1.94·17-s + 0.917·19-s − 0.834·23-s + 3/5·25-s + 1.85·29-s + 0.718·31-s + 1.35·35-s + 0.937·41-s + 2.43·43-s − 0.583·47-s + 1/7·49-s + 1.64·53-s + 1.07·55-s − 1.04·59-s − 1.28·61-s + 0.992·65-s − 1.95·67-s − 1.42·71-s + 1.82·77-s − 2.70·79-s + 1.75·83-s − 1.73·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 47 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 26 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 195 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 203 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−7.70499235356185049090524137752, −7.61212519816430136191225311041, −7.08980046868728208480659429880, −7.08573944995097243391843678326, −6.23719803527863831631338257389, −6.19510100015311410835839138985, −5.59256184134279425860514787242, −5.56223014607784971367768356097, −4.91106773226966831728154889121, −4.55059922427790729891127204867, −4.23960367172535799504270483431, −3.79449599467178956787808186237, −3.13197519027158957337969608459, −3.07065888313650283815987494874, −2.65377021339964261811604639497, −2.46208992163732046267157731898, −1.22767605720874635217944860433, −1.08962072755833382167897291675, 0, 0,
1.08962072755833382167897291675, 1.22767605720874635217944860433, 2.46208992163732046267157731898, 2.65377021339964261811604639497, 3.07065888313650283815987494874, 3.13197519027158957337969608459, 3.79449599467178956787808186237, 4.23960367172535799504270483431, 4.55059922427790729891127204867, 4.91106773226966831728154889121, 5.56223014607784971367768356097, 5.59256184134279425860514787242, 6.19510100015311410835839138985, 6.23719803527863831631338257389, 7.08573944995097243391843678326, 7.08980046868728208480659429880, 7.61212519816430136191225311041, 7.70499235356185049090524137752
