| L(s) = 1 | + 2·5-s − 2·7-s − 6·11-s − 6·13-s + 10·17-s + 2·19-s − 2·23-s − 2·25-s + 8·29-s − 8·31-s − 4·35-s + 4·41-s − 8·43-s − 4·47-s + 3·49-s + 8·53-s − 12·55-s − 16·59-s + 8·61-s − 12·65-s + 6·67-s − 12·71-s + 12·77-s + 18·79-s + 4·83-s + 20·85-s − 4·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.80·11-s − 1.66·13-s + 2.42·17-s + 0.458·19-s − 0.417·23-s − 2/5·25-s + 1.48·29-s − 1.43·31-s − 0.676·35-s + 0.624·41-s − 1.21·43-s − 0.583·47-s + 3/7·49-s + 1.09·53-s − 1.61·55-s − 2.08·59-s + 1.02·61-s − 1.48·65-s + 0.733·67-s − 1.42·71-s + 1.36·77-s + 2.02·79-s + 0.439·83-s + 2.16·85-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 190 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.66372358086910743812149147044, −7.21485225397475412147114786577, −6.74501910984697976146392828346, −6.67153638661439331718921644210, −5.92794714347534419531652616140, −5.79296642442226513235641897268, −5.41620808211978932750062680226, −5.28906491480325043859367010865, −4.82872440384361548174008013631, −4.63827208955850708369141916845, −3.71769567864192478583617791018, −3.71236298188686259298093930437, −3.03080944469916260484713513206, −2.86150048264250425115867151696, −2.37727265390168097453209292553, −2.19628579670689065891934734060, −1.40770261180097262244541817008, −1.06253234533701804970516112836, 0, 0,
1.06253234533701804970516112836, 1.40770261180097262244541817008, 2.19628579670689065891934734060, 2.37727265390168097453209292553, 2.86150048264250425115867151696, 3.03080944469916260484713513206, 3.71236298188686259298093930437, 3.71769567864192478583617791018, 4.63827208955850708369141916845, 4.82872440384361548174008013631, 5.28906491480325043859367010865, 5.41620808211978932750062680226, 5.79296642442226513235641897268, 5.92794714347534419531652616140, 6.67153638661439331718921644210, 6.74501910984697976146392828346, 7.21485225397475412147114786577, 7.66372358086910743812149147044
