| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 4·14-s + 16-s − 6·23-s + 2·25-s + 4·28-s + 7·31-s + 32-s + 3·41-s − 6·46-s + 6·47-s − 2·49-s + 2·50-s + 4·56-s + 7·62-s + 64-s − 9·71-s + 10·73-s − 2·79-s + 3·82-s − 6·92-s + 6·94-s + 4·97-s − 2·98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.06·14-s + 1/4·16-s − 1.25·23-s + 2/5·25-s + 0.755·28-s + 1.25·31-s + 0.176·32-s + 0.468·41-s − 0.884·46-s + 0.875·47-s − 2/7·49-s + 0.282·50-s + 0.534·56-s + 0.889·62-s + 1/8·64-s − 1.06·71-s + 1.17·73-s − 0.225·79-s + 0.331·82-s − 0.625·92-s + 0.618·94-s + 0.406·97-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.805682928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.805682928\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 136 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.615628600082486989893544792085, −9.155973775859180310817956096058, −8.398043395457038003743427800884, −8.059926966783315414331864250492, −7.77391214690864955069370501579, −7.05278046389838384100187795672, −6.51629737012322741969188174032, −5.92883306070050560908107501594, −5.38852230647222417271551747254, −4.80232702965395195965982690461, −4.37965135088388926206575989318, −3.80312942371805057599945851082, −2.86821523080352147795117507023, −2.15098205066474558330636560485, −1.31089221869471524348825772553,
1.31089221869471524348825772553, 2.15098205066474558330636560485, 2.86821523080352147795117507023, 3.80312942371805057599945851082, 4.37965135088388926206575989318, 4.80232702965395195965982690461, 5.38852230647222417271551747254, 5.92883306070050560908107501594, 6.51629737012322741969188174032, 7.05278046389838384100187795672, 7.77391214690864955069370501579, 8.059926966783315414331864250492, 8.398043395457038003743427800884, 9.155973775859180310817956096058, 9.615628600082486989893544792085
