| L(s) = 1 | − 9·3-s + 6·5-s + 108·11-s + 692·13-s − 54·15-s − 1.39e3·17-s − 1.01e3·19-s + 1.53e3·23-s + 3.12e3·25-s + 729·27-s − 7.52e3·29-s − 736·31-s − 972·33-s − 2.05e3·37-s − 6.22e3·39-s + 3.10e4·41-s + 2.20e4·43-s + 4.56e3·47-s + 1.25e4·51-s + 7.96e3·53-s + 648·55-s + 9.10e3·57-s − 7.02e3·59-s + 2.68e4·61-s + 4.15e3·65-s − 5.21e4·67-s − 1.38e4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.107·5-s + 0.269·11-s + 1.13·13-s − 0.0619·15-s − 1.17·17-s − 0.643·19-s + 0.605·23-s + 25-s + 0.192·27-s − 1.66·29-s − 0.137·31-s − 0.155·33-s − 0.246·37-s − 0.655·39-s + 2.88·41-s + 1.82·43-s + 0.301·47-s + 0.677·51-s + 0.389·53-s + 0.0288·55-s + 0.371·57-s − 0.262·59-s + 0.924·61-s + 0.121·65-s − 1.41·67-s − 0.349·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.099724711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099724711\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 3089 T^{2} - 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 108 T - 149387 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 346 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1398 T + 534547 T^{2} + 1398 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1012 T - 1451955 T^{2} + 1012 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1536 T - 4077047 T^{2} - 1536 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3762 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 736 T - 28087455 T^{2} + 736 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2054 T - 65125041 T^{2} + 2054 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 15534 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11036 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4560 T - 208551407 T^{2} - 4560 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7962 T - 354802049 T^{2} - 7962 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7020 T - 665643899 T^{2} + 7020 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26870 T - 122599401 T^{2} - 26870 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 52148 T + 1369288797 T^{2} + 52148 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2544 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9766 T - 1977696837 T^{2} + 9766 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 68672 T + 1638787185 T^{2} + 68672 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 61668 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 41454 T - 3865625333 T^{2} + 41454 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 111262 T + p^{5} T^{2} )^{2} \) |
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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
−10.36698985353765156638547604776, −9.528337389829272732620330177267, −9.190808255259605017982161306798, −8.838015887157904097121141093133, −8.654839616935570264478061488849, −7.76220746827376397213520900871, −7.48629173440886086438693332193, −6.96970175979985399279403389929, −6.39114684761249682878598829368, −6.07664840979882166647606864361, −5.70057542269750718842104996590, −5.12108603559051125598259846840, −4.49329710887603635966083989687, −4.03257380954864532415924348813, −3.69272304855845498282940634837, −2.70714690507075723057389303145, −2.40641408572970467990534833497, −1.54549986769684891977316426019, −0.979468746839186919338525307860, −0.39018407369539893220679879472,
0.39018407369539893220679879472, 0.979468746839186919338525307860, 1.54549986769684891977316426019, 2.40641408572970467990534833497, 2.70714690507075723057389303145, 3.69272304855845498282940634837, 4.03257380954864532415924348813, 4.49329710887603635966083989687, 5.12108603559051125598259846840, 5.70057542269750718842104996590, 6.07664840979882166647606864361, 6.39114684761249682878598829368, 6.96970175979985399279403389929, 7.48629173440886086438693332193, 7.76220746827376397213520900871, 8.654839616935570264478061488849, 8.838015887157904097121141093133, 9.190808255259605017982161306798, 9.528337389829272732620330177267, 10.36698985353765156638547604776
