| L(s) = 1 | + 2·5-s − 2·7-s + 11-s − 7·17-s + 2·19-s + 8·23-s − 7·25-s + 5·29-s − 13·31-s − 4·35-s + 3·41-s + 8·47-s + 3·49-s − 3·53-s + 2·55-s − 4·59-s + 2·61-s − 11·67-s + 4·71-s − 73-s − 2·77-s − 28·79-s + 3·83-s − 14·85-s − 6·89-s + 4·95-s + 12·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.301·11-s − 1.69·17-s + 0.458·19-s + 1.66·23-s − 7/5·25-s + 0.928·29-s − 2.33·31-s − 0.676·35-s + 0.468·41-s + 1.16·47-s + 3/7·49-s − 0.412·53-s + 0.269·55-s − 0.520·59-s + 0.256·61-s − 1.34·67-s + 0.474·71-s − 0.117·73-s − 0.227·77-s − 3.15·79-s + 0.329·83-s − 1.51·85-s − 0.635·89-s + 0.410·95-s + 1.21·97-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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 49 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 61 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 13 T + 101 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 81 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 109 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 71 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 135 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 29 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 117 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 87 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 113 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 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
−7.29354640700253338997796075671, −7.19054524223103678107597163486, −6.72028562627675005826315610034, −6.68324536357205240430201925708, −6.01296835659468474592498428011, −5.91543648698521149769934500526, −5.45376046643810630949704957623, −5.34133344286343828268205149554, −4.62784896402695750351576166627, −4.46481893011558096818479999483, −3.93152066381406365090840730093, −3.73922481929958398787677790064, −3.04905394863737767405049431209, −2.90361156975099839621691011115, −2.24282487470900408554313121451, −2.16652981285852268534514010607, −1.28931383978205027998475117348, −1.26971681350274720262374824784, 0, 0,
1.26971681350274720262374824784, 1.28931383978205027998475117348, 2.16652981285852268534514010607, 2.24282487470900408554313121451, 2.90361156975099839621691011115, 3.04905394863737767405049431209, 3.73922481929958398787677790064, 3.93152066381406365090840730093, 4.46481893011558096818479999483, 4.62784896402695750351576166627, 5.34133344286343828268205149554, 5.45376046643810630949704957623, 5.91543648698521149769934500526, 6.01296835659468474592498428011, 6.68324536357205240430201925708, 6.72028562627675005826315610034, 7.19054524223103678107597163486, 7.29354640700253338997796075671
