| L(s) = 1 | − 3-s + 8·7-s + 9-s − 8·13-s + 2·19-s − 8·21-s + 6·25-s − 27-s + 4·31-s + 8·37-s + 8·39-s + 8·43-s + 34·49-s − 2·57-s − 20·61-s + 8·63-s + 16·67-s − 4·73-s − 6·75-s + 28·79-s + 81-s − 64·91-s − 4·93-s + 28·97-s − 4·103-s − 32·109-s − 8·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3.02·7-s + 1/3·9-s − 2.21·13-s + 0.458·19-s − 1.74·21-s + 6/5·25-s − 0.192·27-s + 0.718·31-s + 1.31·37-s + 1.28·39-s + 1.21·43-s + 34/7·49-s − 0.264·57-s − 2.56·61-s + 1.00·63-s + 1.95·67-s − 0.468·73-s − 0.692·75-s + 3.15·79-s + 1/9·81-s − 6.70·91-s − 0.414·93-s + 2.84·97-s − 0.394·103-s − 3.06·109-s − 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.348289350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.348289350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.213102607783853972181362930470, −7.83209972276749201593977871816, −7.47928872381719998670522703456, −7.39148933340994032193671942533, −6.56715591626159015914612137952, −6.08417078961404314892413228818, −5.32267004942502611581924636194, −4.98783298689795603138187378693, −4.80382850580496834483645089906, −4.50335507641114582801242921882, −3.76647440070571202785776880394, −2.58336626686734098148346747903, −2.38466653285533258222457062316, −1.53184510895126931162943066328, −0.876714004684985756350914094634,
0.876714004684985756350914094634, 1.53184510895126931162943066328, 2.38466653285533258222457062316, 2.58336626686734098148346747903, 3.76647440070571202785776880394, 4.50335507641114582801242921882, 4.80382850580496834483645089906, 4.98783298689795603138187378693, 5.32267004942502611581924636194, 6.08417078961404314892413228818, 6.56715591626159015914612137952, 7.39148933340994032193671942533, 7.47928872381719998670522703456, 7.83209972276749201593977871816, 8.213102607783853972181362930470
