| L(s) = 1 | + 5-s − 11-s − 4·13-s − 3·17-s + 5·19-s − 3·23-s + 5·25-s + 12·29-s − 31-s + 5·37-s − 20·41-s − 8·43-s − 47-s − 9·53-s − 55-s − 3·59-s + 3·61-s − 4·65-s − 11·67-s − 32·71-s + 7·73-s + 11·79-s − 8·83-s − 3·85-s + 9·89-s + 5·95-s − 12·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.625·23-s + 25-s + 2.22·29-s − 0.179·31-s + 0.821·37-s − 3.12·41-s − 1.21·43-s − 0.145·47-s − 1.23·53-s − 0.134·55-s − 0.390·59-s + 0.384·61-s − 0.496·65-s − 1.34·67-s − 3.79·71-s + 0.819·73-s + 1.23·79-s − 0.878·83-s − 0.325·85-s + 0.953·89-s + 0.512·95-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233796529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233796529\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 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
−8.677264777718067008888600378818, −8.384801328355182377862302508174, −8.166423310383080954810556587018, −7.50869874515317231007050691188, −7.31726766036183524478256079266, −6.90607958541045003457299432342, −6.56202825699481683822786721348, −6.05568213534880117769942171714, −5.97523649418232971322541172816, −5.03798883183762878239337995928, −4.97960367384078813936593535257, −4.79613761986216462137133710896, −4.29600244192821766558779229927, −3.48890427283877296543872776650, −3.23116615740893548146609606076, −2.71244141787279812490716823366, −2.43213559524387488486262845474, −1.60678278311467160890563830409, −1.36663520754116046331529596047, −0.32665185366113103000616308488,
0.32665185366113103000616308488, 1.36663520754116046331529596047, 1.60678278311467160890563830409, 2.43213559524387488486262845474, 2.71244141787279812490716823366, 3.23116615740893548146609606076, 3.48890427283877296543872776650, 4.29600244192821766558779229927, 4.79613761986216462137133710896, 4.97960367384078813936593535257, 5.03798883183762878239337995928, 5.97523649418232971322541172816, 6.05568213534880117769942171714, 6.56202825699481683822786721348, 6.90607958541045003457299432342, 7.31726766036183524478256079266, 7.50869874515317231007050691188, 8.166423310383080954810556587018, 8.384801328355182377862302508174, 8.677264777718067008888600378818
