| L(s) = 1 | − 3-s + 5-s + 5·7-s + 3·9-s − 6·11-s + 4·13-s − 15-s + 6·17-s − 8·19-s − 5·21-s − 3·23-s − 8·27-s + 6·29-s − 2·31-s + 6·33-s + 5·35-s − 8·37-s − 4·39-s − 6·41-s + 10·43-s + 3·45-s + 18·49-s − 6·51-s − 12·53-s − 6·55-s + 8·57-s + 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.88·7-s + 9-s − 1.80·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s − 1.09·21-s − 0.625·23-s − 1.53·27-s + 1.11·29-s − 0.359·31-s + 1.04·33-s + 0.845·35-s − 1.31·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.447·45-s + 18/7·49-s − 0.840·51-s − 1.64·53-s − 0.809·55-s + 1.05·57-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223974200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223974200\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−13.36706048412841757629063522976, −12.92537522565612462755954713376, −12.22731696367484953935176051160, −12.12519234019566390850642585439, −11.08619625972759325409613725196, −10.90730112731691485270055581781, −10.49487441071752086039410885698, −10.10354980205228988436042381203, −9.338167679084910960978160382074, −8.444227188983535485232190633415, −8.064615327217175668888713984766, −7.82713327274107256128074261923, −6.99365084362842027402329299925, −6.20864127638802650703356683793, −5.53064951770157521542506418499, −5.17040697582510466059222722227, −4.48615001777838913573719204557, −3.72731121181989691548103376156, −2.30107921941669512543073014083, −1.50858264136031691399360890224,
1.50858264136031691399360890224, 2.30107921941669512543073014083, 3.72731121181989691548103376156, 4.48615001777838913573719204557, 5.17040697582510466059222722227, 5.53064951770157521542506418499, 6.20864127638802650703356683793, 6.99365084362842027402329299925, 7.82713327274107256128074261923, 8.064615327217175668888713984766, 8.444227188983535485232190633415, 9.338167679084910960978160382074, 10.10354980205228988436042381203, 10.49487441071752086039410885698, 10.90730112731691485270055581781, 11.08619625972759325409613725196, 12.12519234019566390850642585439, 12.22731696367484953935176051160, 12.92537522565612462755954713376, 13.36706048412841757629063522976
