| L(s) = 1 | + 31·3-s − 64·4-s + 718·9-s − 1.98e3·12-s + 3.07e3·16-s − 3.00e3·25-s + 1.47e4·27-s + 1.55e4·31-s − 4.59e4·36-s − 2.53e3·37-s + 9.52e4·48-s + 3.36e4·49-s − 1.31e5·64-s + 1.45e5·67-s − 9.30e4·75-s + 2.82e5·81-s + 4.82e5·93-s − 3.26e5·97-s + 1.92e5·100-s − 3.60e5·103-s − 9.42e5·108-s − 7.85e4·111-s − 1.61e5·121-s − 9.95e5·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.98·3-s − 2·4-s + 2.95·9-s − 3.97·12-s + 3·16-s − 0.960·25-s + 3.88·27-s + 2.90·31-s − 5.90·36-s − 0.304·37-s + 5.96·48-s + 2·49-s − 4·64-s + 3.96·67-s − 1.90·75-s + 4.77·81-s + 5.77·93-s − 3.52·97-s + 1.92·100-s − 3.34·103-s − 7.77·108-s − 0.605·111-s − 121-s − 5.81·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.677060916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677060916\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - 31 T + p^{5} T^{2} \) |
| 11 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 57 T + p^{5} T^{2} )( 1 + 57 T + p^{5} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 981 T + p^{5} T^{2} )( 1 + 981 T + p^{5} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7775 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1267 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 24708 T + p^{5} T^{2} )( 1 + 24708 T + p^{5} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 34806 T + p^{5} T^{2} )( 1 + 34806 T + p^{5} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 24825 T + p^{5} T^{2} )( 1 + 24825 T + p^{5} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 72917 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 66273 T + p^{5} T^{2} )( 1 + 66273 T + p^{5} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 91089 T + p^{5} T^{2} )( 1 + 91089 T + p^{5} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 163183 T + p^{5} T^{2} )^{2} \) |
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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
−15.46586949043423593480956028835, −15.40478654736815424023554686792, −14.54788478364210083544831260402, −13.97790261706949847000263338902, −13.71817566051276078337238037839, −13.36704896334722217577147069374, −12.57658501569037454899175396393, −12.11955364246724241460641633706, −10.53635798038715966675450442312, −9.725057013617993165734644998908, −9.685361917310254111506945061205, −8.817082561242897380721959415442, −8.146454866320718588056049499674, −8.050547313759792387808329745339, −6.79509751801134785186566336813, −5.29633726447418472735087100040, −4.27470267460089352604975364464, −3.85653339879809286231674913238, −2.69402943737463911088975892599, −1.02269945630030399861323612426,
1.02269945630030399861323612426, 2.69402943737463911088975892599, 3.85653339879809286231674913238, 4.27470267460089352604975364464, 5.29633726447418472735087100040, 6.79509751801134785186566336813, 8.050547313759792387808329745339, 8.146454866320718588056049499674, 8.817082561242897380721959415442, 9.685361917310254111506945061205, 9.725057013617993165734644998908, 10.53635798038715966675450442312, 12.11955364246724241460641633706, 12.57658501569037454899175396393, 13.36704896334722217577147069374, 13.71817566051276078337238037839, 13.97790261706949847000263338902, 14.54788478364210083544831260402, 15.40478654736815424023554686792, 15.46586949043423593480956028835
