| L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 8·13-s − 4·16-s + 4·21-s − 5·27-s + 14·31-s − 6·37-s − 8·39-s + 12·43-s − 4·48-s − 2·49-s + 24·61-s − 8·63-s + 14·67-s − 8·73-s − 20·79-s + 81-s − 32·91-s + 14·93-s + 14·97-s + 32·103-s + 20·109-s − 6·111-s − 16·112-s + 16·117-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 2.21·13-s − 16-s + 0.872·21-s − 0.962·27-s + 2.51·31-s − 0.986·37-s − 1.28·39-s + 1.82·43-s − 0.577·48-s − 2/7·49-s + 3.07·61-s − 1.00·63-s + 1.71·67-s − 0.936·73-s − 2.25·79-s + 1/9·81-s − 3.35·91-s + 1.45·93-s + 1.42·97-s + 3.15·103-s + 1.91·109-s − 0.569·111-s − 1.51·112-s + 1.47·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.137978418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137978418\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−8.276530745833509741483457469247, −7.952532350978681778925001629133, −7.56082953278459650643638402038, −7.06748750449497583777914546446, −6.74210091587050441798111318378, −5.99803243316135624496921094343, −5.49322091026488161466653442724, −4.90502322677283757606823458951, −4.70136976599285716284507090528, −4.29108769550776490491760393895, −3.45386803680567664069181238677, −2.69851752539611099704924785810, −2.35074848930377289705338533775, −1.91502190096627856445198269376, −0.68132836611621051670216891055,
0.68132836611621051670216891055, 1.91502190096627856445198269376, 2.35074848930377289705338533775, 2.69851752539611099704924785810, 3.45386803680567664069181238677, 4.29108769550776490491760393895, 4.70136976599285716284507090528, 4.90502322677283757606823458951, 5.49322091026488161466653442724, 5.99803243316135624496921094343, 6.74210091587050441798111318378, 7.06748750449497583777914546446, 7.56082953278459650643638402038, 7.952532350978681778925001629133, 8.276530745833509741483457469247
