L(s) = 1 | + 2·5-s − 2·7-s − 2·17-s + 2·19-s − 8·23-s − 2·25-s − 10·29-s + 4·31-s − 4·35-s + 4·41-s + 16·43-s + 2·47-s + 3·49-s + 2·53-s + 8·59-s − 16·61-s − 12·67-s − 6·71-s − 24·79-s − 2·83-s − 4·85-s − 4·89-s + 4·95-s + 16·97-s − 14·101-s + 2·107-s + 12·109-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.485·17-s + 0.458·19-s − 1.66·23-s − 2/5·25-s − 1.85·29-s + 0.718·31-s − 0.676·35-s + 0.624·41-s + 2.43·43-s + 0.291·47-s + 3/7·49-s + 0.274·53-s + 1.04·59-s − 2.04·61-s − 1.46·67-s − 0.712·71-s − 2.70·79-s − 0.219·83-s − 0.433·85-s − 0.423·89-s + 0.410·95-s + 1.62·97-s − 1.39·101-s + 0.193·107-s + 1.14·109-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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 122 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.41899486715441144324200888191, −7.31632370874229010607916925332, −6.81734263652294456454082607283, −6.36711393436554598551836789273, −5.98090463534957735751113656080, −5.95676842864901874450457831269, −5.58549832141792829673213412358, −5.34994363652599251282863156397, −4.62231608038667603239056930205, −4.35195028037816390684284027779, −3.93676998670209969985001390485, −3.81719112521580628574694423564, −3.09453400936313081601225809344, −2.81445499711354156946386247002, −2.21713228050150796281853352126, −2.21113185854446894071579297139, −1.40704538397475749011271734054, −1.15104211008351401609105827360, 0, 0,
1.15104211008351401609105827360, 1.40704538397475749011271734054, 2.21113185854446894071579297139, 2.21713228050150796281853352126, 2.81445499711354156946386247002, 3.09453400936313081601225809344, 3.81719112521580628574694423564, 3.93676998670209969985001390485, 4.35195028037816390684284027779, 4.62231608038667603239056930205, 5.34994363652599251282863156397, 5.58549832141792829673213412358, 5.95676842864901874450457831269, 5.98090463534957735751113656080, 6.36711393436554598551836789273, 6.81734263652294456454082607283, 7.31632370874229010607916925332, 7.41899486715441144324200888191