| L(s) = 1 | + 2·5-s + 3·9-s + 4·11-s − 4·13-s − 6·17-s + 8·19-s + 5·25-s + 12·29-s + 8·31-s + 2·37-s − 4·41-s − 8·43-s + 6·45-s − 8·47-s − 6·53-s + 8·55-s − 6·61-s − 8·65-s + 4·67-s − 16·71-s + 10·73-s − 16·79-s − 16·83-s − 12·85-s − 6·89-s + 16·95-s + 12·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 9-s + 1.20·11-s − 1.10·13-s − 1.45·17-s + 1.83·19-s + 25-s + 2.22·29-s + 1.43·31-s + 0.328·37-s − 0.624·41-s − 1.21·43-s + 0.894·45-s − 1.16·47-s − 0.824·53-s + 1.07·55-s − 0.768·61-s − 0.992·65-s + 0.488·67-s − 1.89·71-s + 1.17·73-s − 1.80·79-s − 1.75·83-s − 1.30·85-s − 0.635·89-s + 1.64·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.259680167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259680167\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−11.40344511286972910858701886830, −11.32043421468784366133231299063, −10.36785676688309124009573448310, −9.929124609426543570709263219077, −9.917274417527311841030956296240, −9.439663427729552651315334271770, −8.725288688985365401154026504464, −8.539230659751575763907655294321, −7.76044616594718079628312705115, −7.14589684481559570253666464729, −6.71110751894114905772863100706, −6.54780586936613037623635497675, −5.86187912253282319375876365814, −5.06525128929975497774589015480, −4.52405899190915212718345758041, −4.46263616498577605798171719832, −3.19527635429439106026589972181, −2.82041761037243572874998032881, −1.80138803823286742340519672064, −1.15134921379481204045332394617,
1.15134921379481204045332394617, 1.80138803823286742340519672064, 2.82041761037243572874998032881, 3.19527635429439106026589972181, 4.46263616498577605798171719832, 4.52405899190915212718345758041, 5.06525128929975497774589015480, 5.86187912253282319375876365814, 6.54780586936613037623635497675, 6.71110751894114905772863100706, 7.14589684481559570253666464729, 7.76044616594718079628312705115, 8.539230659751575763907655294321, 8.725288688985365401154026504464, 9.439663427729552651315334271770, 9.917274417527311841030956296240, 9.929124609426543570709263219077, 10.36785676688309124009573448310, 11.32043421468784366133231299063, 11.40344511286972910858701886830
