L(s) = 1 | + 3-s + 6·7-s + 9-s − 2·13-s − 4·16-s − 10·19-s + 6·21-s + 27-s − 6·31-s − 4·37-s − 2·39-s − 2·43-s − 4·48-s + 13·49-s − 10·57-s + 14·61-s + 6·63-s + 6·67-s + 28·73-s + 81-s − 12·91-s − 6·93-s − 34·97-s + 8·103-s + 10·109-s − 4·111-s − 24·112-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 2.26·7-s + 1/3·9-s − 0.554·13-s − 16-s − 2.29·19-s + 1.30·21-s + 0.192·27-s − 1.07·31-s − 0.657·37-s − 0.320·39-s − 0.304·43-s − 0.577·48-s + 13/7·49-s − 1.32·57-s + 1.79·61-s + 0.755·63-s + 0.733·67-s + 3.27·73-s + 1/9·81-s − 1.25·91-s − 0.622·93-s − 3.45·97-s + 0.788·103-s + 0.957·109-s − 0.379·111-s − 2.26·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.512474380\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.512474380\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−10.88974094536361104567387978062, −10.84373752796776504705892968641, −9.903619965777680376182050580499, −9.294135845295806835405725735271, −8.655074250265874594654667643843, −8.114757177440921419069207299350, −8.102126663118456650752904190012, −6.99182605075727565742724929802, −6.77287251257420576381216593643, −5.60694328736998842121660037613, −4.92213034669592094010306499374, −4.48018036005395779311676580836, −3.78545707329788062312381068333, −2.27079729708413704097834802658, −1.90319955654708104595670502735,
1.90319955654708104595670502735, 2.27079729708413704097834802658, 3.78545707329788062312381068333, 4.48018036005395779311676580836, 4.92213034669592094010306499374, 5.60694328736998842121660037613, 6.77287251257420576381216593643, 6.99182605075727565742724929802, 8.102126663118456650752904190012, 8.114757177440921419069207299350, 8.655074250265874594654667643843, 9.294135845295806835405725735271, 9.903619965777680376182050580499, 10.84373752796776504705892968641, 10.88974094536361104567387978062