| L(s) = 1 | − 2-s + 4-s − 8-s + 2·9-s + 16-s − 2·18-s + 2·23-s + 4·25-s − 32-s + 2·36-s + 16·41-s − 2·46-s − 10·49-s − 4·50-s + 64-s − 6·71-s − 2·72-s + 10·73-s + 8·79-s − 5·81-s − 16·82-s + 18·89-s + 2·92-s − 8·97-s + 10·98-s + 4·100-s + 16·103-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1/4·16-s − 0.471·18-s + 0.417·23-s + 4/5·25-s − 0.176·32-s + 1/3·36-s + 2.49·41-s − 0.294·46-s − 1.42·49-s − 0.565·50-s + 1/8·64-s − 0.712·71-s − 0.235·72-s + 1.17·73-s + 0.900·79-s − 5/9·81-s − 1.76·82-s + 1.90·89-s + 0.208·92-s − 0.812·97-s + 1.01·98-s + 2/5·100-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.023824691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023824691\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 337 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 22 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−10.17616853429269143746639228177, −9.693533140244474417410851110009, −9.125998497611898163672319767116, −8.859683185429454891494124521203, −8.070942408086148461775812363022, −7.62759273933525697228137594905, −7.20556230811789578834340889471, −6.48080127294790585219433343440, −6.13824516864621888525406997654, −5.23801452913251368952217695361, −4.66032191614609348096878146118, −3.89254302586780986899300212867, −3.05424944097476422628487586978, −2.20424260853250243646758127252, −1.07828846811749494375065134925,
1.07828846811749494375065134925, 2.20424260853250243646758127252, 3.05424944097476422628487586978, 3.89254302586780986899300212867, 4.66032191614609348096878146118, 5.23801452913251368952217695361, 6.13824516864621888525406997654, 6.48080127294790585219433343440, 7.20556230811789578834340889471, 7.62759273933525697228137594905, 8.070942408086148461775812363022, 8.859683185429454891494124521203, 9.125998497611898163672319767116, 9.693533140244474417410851110009, 10.17616853429269143746639228177
