L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s − 3·9-s − 2·11-s + 4·15-s − 4·17-s + 6·19-s + 4·21-s − 2·23-s + 3·25-s + 14·27-s + 6·29-s + 4·31-s + 4·33-s + 4·35-s + 14·37-s − 2·41-s − 2·43-s + 6·45-s − 6·47-s − 4·49-s + 8·51-s + 4·55-s − 12·57-s + 22·61-s + 6·63-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s + 1.03·15-s − 0.970·17-s + 1.37·19-s + 0.872·21-s − 0.417·23-s + 3/5·25-s + 2.69·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s + 0.676·35-s + 2.30·37-s − 0.312·41-s − 0.304·43-s + 0.894·45-s − 0.875·47-s − 4/7·49-s + 1.12·51-s + 0.539·55-s − 1.58·57-s + 2.81·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 22 T + 236 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 151 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 143 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 152 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 172 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.57421918920739435823552501649, −7.55306465633923667146324093196, −6.82566122908499315938674769132, −6.63208801706408142281533666958, −6.24717491994839372136430377603, −6.18466125187490095604837787441, −5.49107043326671440940537069844, −5.40005367423198638552069874833, −4.90079111112875617175351993331, −4.68790677975834860749592655280, −4.20438359721468250907638695852, −3.77960757537727581972141320797, −3.17082466691641110541326345160, −3.05533237109292968820396085508, −2.51130969686834949481156066336, −2.31611668507131549555156728291, −1.10812026230204212599112130695, −0.912552782890851832960504696305, 0, 0,
0.912552782890851832960504696305, 1.10812026230204212599112130695, 2.31611668507131549555156728291, 2.51130969686834949481156066336, 3.05533237109292968820396085508, 3.17082466691641110541326345160, 3.77960757537727581972141320797, 4.20438359721468250907638695852, 4.68790677975834860749592655280, 4.90079111112875617175351993331, 5.40005367423198638552069874833, 5.49107043326671440940537069844, 6.18466125187490095604837787441, 6.24717491994839372136430377603, 6.63208801706408142281533666958, 6.82566122908499315938674769132, 7.55306465633923667146324093196, 7.57421918920739435823552501649