| L(s) = 1 | + 2-s + 4-s + 8-s + 2·9-s + 16-s + 2·18-s + 4·23-s − 2·25-s + 8·31-s + 32-s + 2·36-s − 16·41-s + 4·46-s + 8·47-s − 10·49-s − 2·50-s + 8·62-s + 64-s + 4·71-s + 2·72-s + 4·73-s + 4·79-s − 5·81-s − 16·82-s + 12·89-s + 4·92-s + 8·94-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 + 0.834·23-s − 2/5·25-s + 1.43·31-s + 0.176·32-s + 1/3·36-s − 2.49·41-s + 0.589·46-s + 1.16·47-s − 1.42·49-s − 0.282·50-s + 1.01·62-s + 1/8·64-s + 0.474·71-s + 0.235·72-s + 0.468·73-s + 0.450·79-s − 5/9·81-s − 1.76·82-s + 1.27·89-s + 0.417·92-s + 0.825·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086025662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086025662\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−10.38291537136051732969141257352, −9.902259407270832677670593609756, −9.451296141810670386954799840607, −8.695753639094576479572487695577, −8.189891719428137489932720998971, −7.64590483359088144833483750233, −6.91258782760502726837506572186, −6.64137900699635468792817893624, −5.97186184552616425240703115851, −5.13703379951568419997042555919, −4.80598764548361421890911961586, −4.01258806677672247500940815499, −3.35841613549389650401246837639, −2.52107048876082081684395403964, −1.43624842265480697539962573664,
1.43624842265480697539962573664, 2.52107048876082081684395403964, 3.35841613549389650401246837639, 4.01258806677672247500940815499, 4.80598764548361421890911961586, 5.13703379951568419997042555919, 5.97186184552616425240703115851, 6.64137900699635468792817893624, 6.91258782760502726837506572186, 7.64590483359088144833483750233, 8.189891719428137489932720998971, 8.695753639094576479572487695577, 9.451296141810670386954799840607, 9.902259407270832677670593609756, 10.38291537136051732969141257352
