| L(s) = 1 | + 3-s − 2·7-s + 9-s + 12·13-s − 8·19-s − 2·21-s − 6·25-s + 27-s + 16·31-s + 12·37-s + 12·39-s + 8·43-s + 3·49-s − 8·57-s − 4·61-s − 2·63-s − 24·67-s − 28·73-s − 6·75-s + 16·79-s + 81-s − 24·91-s + 16·93-s + 20·97-s − 4·109-s + 12·111-s + 12·117-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 3.32·13-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.192·27-s + 2.87·31-s + 1.97·37-s + 1.92·39-s + 1.21·43-s + 3/7·49-s − 1.05·57-s − 0.512·61-s − 0.251·63-s − 2.93·67-s − 3.27·73-s − 0.692·75-s + 1.80·79-s + 1/9·81-s − 2.51·91-s + 1.65·93-s + 2.03·97-s − 0.383·109-s + 1.13·111-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.409026097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.409026097\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.905190686502749891901965349669, −8.334199259994987818549291908705, −7.917190719920927542416951015280, −7.60026307977982948167798436547, −6.54428519689900750892915035611, −6.43798012578301106351522548543, −5.90970829345295627546285462325, −5.90015243373728427076887013140, −4.40034147894441213970326651837, −4.35622436888448856514855173618, −3.82931598942638938100602509659, −3.10934300001651305626809863177, −2.67493504075964250922211563542, −1.70572377211895013061556457420, −0.925974150930484580155748442830,
0.925974150930484580155748442830, 1.70572377211895013061556457420, 2.67493504075964250922211563542, 3.10934300001651305626809863177, 3.82931598942638938100602509659, 4.35622436888448856514855173618, 4.40034147894441213970326651837, 5.90015243373728427076887013140, 5.90970829345295627546285462325, 6.43798012578301106351522548543, 6.54428519689900750892915035611, 7.60026307977982948167798436547, 7.917190719920927542416951015280, 8.334199259994987818549291908705, 8.905190686502749891901965349669
