| L(s) = 1 | − 23·7-s − 191·13-s + 1.29e3·19-s − 625·25-s − 194·31-s + 5.18e3·37-s + 3.21e3·43-s + 2.40e3·49-s + 5.23e3·61-s + 8.80e3·67-s + 1.95e4·73-s + 1.23e4·79-s + 4.39e3·91-s − 9.74e3·97-s − 3.43e3·103-s + 4.40e4·109-s − 1.46e4·121-s + 127-s + 131-s − 2.97e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.469·7-s − 1.13·13-s + 3.58·19-s − 25-s − 0.201·31-s + 3.78·37-s + 1.73·43-s + 49-s + 1.40·61-s + 1.96·67-s + 3.67·73-s + 1.98·79-s + 0.530·91-s − 1.03·97-s − 0.323·103-s + 3.70·109-s − 121-s + 6.20e−5·127-s + 5.82e−5·131-s − 1.68·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.457701831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457701831\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 71 T + p^{4} T^{2} )( 1 + 94 T + p^{4} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 146 T + p^{4} T^{2} )( 1 + 337 T + p^{4} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 647 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 1559 T + p^{4} T^{2} )( 1 + 1753 T + p^{4} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2591 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 3191 T + p^{4} T^{2} )( 1 - 23 T + p^{4} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7199 T + p^{4} T^{2} )( 1 + 1966 T + p^{4} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5906 T + p^{4} T^{2} )( 1 - 2903 T + p^{4} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9791 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 7682 T + p^{4} T^{2} )( 1 - 4679 T + p^{4} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9071 T + p^{4} T^{2} )( 1 + 18814 T + p^{4} T^{2} ) \) |
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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
−11.12177847785443696740505880072, −11.00805170551163166664566939325, −9.860322905715721756821036967348, −9.844665600494469759064933175680, −9.410513765090091291423950710267, −9.276426029516073399808786000862, −8.031590562615429078988023600050, −7.973368198120343489119614863787, −7.25958048298423462165230775750, −7.19605100137889176665628576847, −6.18040194328737237664020213915, −5.87272240618940916972546492789, −5.12950135553836154172595099068, −4.93154076845340244045882595047, −3.84557459404577389657553353586, −3.60321538832587221365575747731, −2.49305937577561699623507325351, −2.48518751708662994692253580319, −0.899484892134494880909182299002, −0.78441914411268765061571758560,
0.78441914411268765061571758560, 0.899484892134494880909182299002, 2.48518751708662994692253580319, 2.49305937577561699623507325351, 3.60321538832587221365575747731, 3.84557459404577389657553353586, 4.93154076845340244045882595047, 5.12950135553836154172595099068, 5.87272240618940916972546492789, 6.18040194328737237664020213915, 7.19605100137889176665628576847, 7.25958048298423462165230775750, 7.973368198120343489119614863787, 8.031590562615429078988023600050, 9.276426029516073399808786000862, 9.410513765090091291423950710267, 9.844665600494469759064933175680, 9.860322905715721756821036967348, 11.00805170551163166664566939325, 11.12177847785443696740505880072
