| L(s) = 1 | − 3-s + 2·7-s + 9-s + 6·19-s − 2·21-s − 2·25-s − 27-s + 4·31-s + 4·37-s − 2·43-s − 3·49-s − 6·57-s + 2·61-s + 2·63-s + 16·67-s + 4·73-s + 2·75-s + 4·79-s + 81-s − 4·93-s − 28·97-s − 8·103-s − 12·109-s − 4·111-s − 4·121-s + 127-s + 2·129-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.37·19-s − 0.436·21-s − 2/5·25-s − 0.192·27-s + 0.718·31-s + 0.657·37-s − 0.304·43-s − 3/7·49-s − 0.794·57-s + 0.256·61-s + 0.251·63-s + 1.95·67-s + 0.468·73-s + 0.230·75-s + 0.450·79-s + 1/9·81-s − 0.414·93-s − 2.84·97-s − 0.788·103-s − 1.14·109-s − 0.379·111-s − 0.363·121-s + 0.0887·127-s + 0.176·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.073737041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073737041\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 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
−11.03689707133494805317142061732, −10.29592664785567612656881243758, −9.694854775784094639828679342847, −9.463326567496842712500019211654, −8.530363486299904866701059875888, −8.044798005797339536267492189660, −7.58028861214898877251621559922, −6.86634394617748145784756264081, −6.33174758392500801681126444093, −5.47778937909077083948613968465, −5.14585891639350060862135775783, −4.37830631932726407379099575568, −3.59515561423226742913559575582, −2.52949676136981922055970652309, −1.27434937581930657134057324625,
1.27434937581930657134057324625, 2.52949676136981922055970652309, 3.59515561423226742913559575582, 4.37830631932726407379099575568, 5.14585891639350060862135775783, 5.47778937909077083948613968465, 6.33174758392500801681126444093, 6.86634394617748145784756264081, 7.58028861214898877251621559922, 8.044798005797339536267492189660, 8.530363486299904866701059875888, 9.463326567496842712500019211654, 9.694854775784094639828679342847, 10.29592664785567612656881243758, 11.03689707133494805317142061732
