| L(s) = 1 | − 3-s + 9-s − 8·13-s + 8·23-s − 2·25-s − 27-s + 16·37-s + 8·39-s − 8·47-s − 2·49-s − 24·59-s − 8·69-s + 8·71-s + 12·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s − 8·107-s + 8·109-s − 16·111-s − 8·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 2.21·13-s + 1.66·23-s − 2/5·25-s − 0.192·27-s + 2.63·37-s + 1.28·39-s − 1.16·47-s − 2/7·49-s − 3.12·59-s − 0.963·69-s + 0.949·71-s + 1.40·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s − 0.773·107-s + 0.766·109-s − 1.51·111-s − 0.739·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.149649513278736098518409003937, −7.82788846904929832141176658903, −7.49108835275199215983548325277, −6.94629876071008359699574942451, −6.56673687003235705281236550164, −6.07228752987419708312778497748, −5.41099623626807820253202365209, −5.05157574470163367911294253727, −4.54638093787446600235305540099, −4.25522841418634214886309908318, −3.19684028078983961348389043731, −2.79516533386473519101187117962, −2.12183395880720807825002471165, −1.14406755381780253051685082157, 0,
1.14406755381780253051685082157, 2.12183395880720807825002471165, 2.79516533386473519101187117962, 3.19684028078983961348389043731, 4.25522841418634214886309908318, 4.54638093787446600235305540099, 5.05157574470163367911294253727, 5.41099623626807820253202365209, 6.07228752987419708312778497748, 6.56673687003235705281236550164, 6.94629876071008359699574942451, 7.49108835275199215983548325277, 7.82788846904929832141176658903, 8.149649513278736098518409003937
