| L(s) = 1 | + 32·4-s − 211·7-s − 854·13-s − 3.14e3·19-s + 3.12e3·25-s − 6.75e3·28-s − 2.72e3·31-s + 6.66e3·37-s + 4.49e4·43-s + 2.77e4·49-s − 2.73e4·52-s + 3.86e4·61-s − 3.27e4·64-s + 3.79e4·67-s + 7.81e4·73-s − 1.00e5·76-s − 9.08e4·79-s + 1.80e5·91-s − 2.68e5·97-s + 1.00e5·100-s + 2.11e5·103-s + 2.47e5·109-s + 1.61e5·121-s − 8.71e4·124-s + 127-s + 131-s + 6.63e5·133-s + ⋯ |
| L(s) = 1 | + 4-s − 1.62·7-s − 1.40·13-s − 1.99·19-s + 25-s − 1.62·28-s − 0.508·31-s + 0.799·37-s + 3.70·43-s + 1.64·49-s − 1.40·52-s + 1.32·61-s − 64-s + 1.03·67-s + 1.71·73-s − 1.99·76-s − 1.63·79-s + 2.28·91-s − 2.90·97-s + 100-s + 1.96·103-s + 1.99·109-s + 121-s − 0.508·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 3.25·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.477713192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477713192\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 211 T + p^{5} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 427 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1432 T + p^{5} T^{2} )( 1 + 1711 T + p^{5} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7601 T + p^{5} T^{2} )( 1 + 10324 T + p^{5} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 16550 T + p^{5} T^{2} )( 1 + 9889 T + p^{5} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22475 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 56927 T + p^{5} T^{2} )( 1 + 18301 T + p^{5} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 73475 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 79577 T + p^{5} T^{2} )( 1 + 1450 T + p^{5} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 9707 T + p^{5} T^{2} )( 1 + 100564 T + p^{5} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 134386 T + p^{5} T^{2} )^{2} \) |
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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
−14.46847659039461493345903431197, −13.64398955816009963221871083143, −12.80087592266306504327133815938, −12.49317080925063326228470202538, −12.41851196854919668044258481653, −11.10283308403451755005909291256, −11.07485335440873072008018116880, −10.21195500788840378869020378486, −9.709342263118265056967428636651, −9.109847853607771610155324182121, −8.416888688048899633377404508713, −7.20800870100179787253572584582, −7.17781922594811372373820198654, −6.29376681186068698486278281416, −5.88768946107084486704374393855, −4.66420969214935304481468916863, −3.81877294048210270237322961338, −2.58941947995594012126329059081, −2.37226808269346661556830159644, −0.53287911063823539317724046957,
0.53287911063823539317724046957, 2.37226808269346661556830159644, 2.58941947995594012126329059081, 3.81877294048210270237322961338, 4.66420969214935304481468916863, 5.88768946107084486704374393855, 6.29376681186068698486278281416, 7.17781922594811372373820198654, 7.20800870100179787253572584582, 8.416888688048899633377404508713, 9.109847853607771610155324182121, 9.709342263118265056967428636651, 10.21195500788840378869020378486, 11.07485335440873072008018116880, 11.10283308403451755005909291256, 12.41851196854919668044258481653, 12.49317080925063326228470202538, 12.80087592266306504327133815938, 13.64398955816009963221871083143, 14.46847659039461493345903431197
