| L(s) = 1 | − 2·5-s − 6·7-s − 4·11-s + 13-s − 6·17-s + 8·19-s − 4·23-s + 5·25-s + 2·29-s − 14·31-s + 12·35-s + 2·37-s + 8·41-s + 7·43-s + 8·47-s + 13·49-s − 8·53-s + 8·55-s − 12·59-s − 5·61-s − 2·65-s − 9·67-s + 2·71-s + 15·73-s + 24·77-s + 11·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2.26·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.83·19-s − 0.834·23-s + 25-s + 0.371·29-s − 2.51·31-s + 2.02·35-s + 0.328·37-s + 1.24·41-s + 1.06·43-s + 1.16·47-s + 13/7·49-s − 1.09·53-s + 1.07·55-s − 1.56·59-s − 0.640·61-s − 0.248·65-s − 1.09·67-s + 0.237·71-s + 1.75·73-s + 2.73·77-s + 1.23·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2554359799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2554359799\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.254588465403944963934804538998, −8.784259842871944441195996534683, −8.153649840448480367314574553022, −7.68297041940295429103608388902, −7.58383624909787730661035929684, −7.17833761686351194720418511256, −6.77330127275235764807102755320, −6.36477781932498180868876626302, −5.85158369645341203304735718996, −5.78598780566470540757099855452, −5.14007424707999011979484774012, −4.60653433172373397212413852645, −4.26378358551390660447476156966, −3.63887174160624148002397516597, −3.34086295204750639435431510327, −3.06837802004600262315453712001, −2.51769483551444735275502002796, −2.01060275037618941154206401212, −0.945272041674421326246316822670, −0.19730853559321172433349778748,
0.19730853559321172433349778748, 0.945272041674421326246316822670, 2.01060275037618941154206401212, 2.51769483551444735275502002796, 3.06837802004600262315453712001, 3.34086295204750639435431510327, 3.63887174160624148002397516597, 4.26378358551390660447476156966, 4.60653433172373397212413852645, 5.14007424707999011979484774012, 5.78598780566470540757099855452, 5.85158369645341203304735718996, 6.36477781932498180868876626302, 6.77330127275235764807102755320, 7.17833761686351194720418511256, 7.58383624909787730661035929684, 7.68297041940295429103608388902, 8.153649840448480367314574553022, 8.784259842871944441195996534683, 9.254588465403944963934804538998
