| L(s) = 1 | + 3·3-s + 14·5-s − 10·7-s − 22·11-s − 91·13-s + 42·15-s − 37·17-s + 30·19-s − 30·21-s − 162·23-s − 103·25-s − 27·27-s + 113·29-s − 392·31-s − 66·33-s − 140·35-s − 13·37-s − 273·39-s − 285·41-s − 246·43-s + 924·47-s + 343·49-s − 111·51-s − 1.07e3·53-s − 308·55-s + 90·57-s + 576·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.25·5-s − 0.539·7-s − 0.603·11-s − 1.94·13-s + 0.722·15-s − 0.527·17-s + 0.362·19-s − 0.311·21-s − 1.46·23-s − 0.823·25-s − 0.192·27-s + 0.723·29-s − 2.27·31-s − 0.348·33-s − 0.676·35-s − 0.0577·37-s − 1.12·39-s − 1.08·41-s − 0.872·43-s + 2.86·47-s + 49-s − 0.304·51-s − 2.78·53-s − 0.755·55-s + 0.209·57-s + 1.27·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.004798433323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004798433323\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 10 T - 243 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 p T - 7 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 37 T - 3544 T^{2} + 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T - 5959 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 162 T + 14077 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 113 T - 11620 T^{2} - 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 196 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 13 T - 50484 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 285 T + 12304 T^{2} + 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 246 T - 18991 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 462 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 537 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 576 T + 126397 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 635 T + 176244 T^{2} - 635 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 202 T - 259959 T^{2} - 202 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1086 T + 821485 T^{2} + 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 805 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 194 T - 667333 T^{2} + 194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1202 T + 532131 T^{2} - 1202 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.49950036716132014414515089839, −9.848957615391255238600726546009, −9.840497167331731853830892298358, −9.035840374928631461841546074396, −8.920145023610678355236830526787, −8.317434358778121964977440262825, −7.64130173056891933213640918852, −7.26722523541226245238527692501, −7.15296313469166780666282008677, −6.08469666697605443331302708344, −6.07482470278223436915330007533, −5.27690259441214228639905394875, −5.19920515309984855740661516888, −4.19665883146835978392188446249, −3.91586279084517783116866237641, −2.91606373878424086424647769692, −2.63800195760024291637682358411, −2.00143037833824063966941124340, −1.64007005488730284389196547953, −0.01716246547149166287698131723,
0.01716246547149166287698131723, 1.64007005488730284389196547953, 2.00143037833824063966941124340, 2.63800195760024291637682358411, 2.91606373878424086424647769692, 3.91586279084517783116866237641, 4.19665883146835978392188446249, 5.19920515309984855740661516888, 5.27690259441214228639905394875, 6.07482470278223436915330007533, 6.08469666697605443331302708344, 7.15296313469166780666282008677, 7.26722523541226245238527692501, 7.64130173056891933213640918852, 8.317434358778121964977440262825, 8.920145023610678355236830526787, 9.035840374928631461841546074396, 9.840497167331731853830892298358, 9.848957615391255238600726546009, 10.49950036716132014414515089839
