| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 4·5-s − 4·6-s + 3·9-s + 8·10-s − 4·12-s + 12·13-s − 8·15-s − 4·16-s + 6·18-s + 8·20-s + 11·25-s + 24·26-s − 4·27-s − 16·30-s − 16·31-s − 8·32-s + 6·36-s + 4·37-s − 24·39-s − 16·41-s + 12·43-s + 12·45-s + 8·48-s − 49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 9-s + 2.52·10-s − 1.15·12-s + 3.32·13-s − 2.06·15-s − 16-s + 1.41·18-s + 1.78·20-s + 11/5·25-s + 4.70·26-s − 0.769·27-s − 2.92·30-s − 2.87·31-s − 1.41·32-s + 36-s + 0.657·37-s − 3.84·39-s − 2.49·41-s + 1.82·43-s + 1.78·45-s + 1.15·48-s − 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.814763411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.814763411\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.63855688308263044073699763356, −10.22055564903449068597804489576, −9.623661535681515484334411058219, −9.150034713544318670605475273109, −8.722047973217996408951500765928, −8.624863431129588132276124551128, −7.65494799383293181616186704631, −6.93524927378989757660266026624, −6.74687877622405378223541267975, −6.14640317875450093577799490037, −5.91943038126992156914162973044, −5.58888338268837600996760223866, −5.44113838224845248127704045827, −4.68967901835867317831478671084, −4.16222293778383568350588310650, −3.47741256638108773246855468454, −3.37809013706155145767483498617, −2.17890364115498778427599068916, −1.72965069967063217883851920422, −1.00520541145744263086135747723,
1.00520541145744263086135747723, 1.72965069967063217883851920422, 2.17890364115498778427599068916, 3.37809013706155145767483498617, 3.47741256638108773246855468454, 4.16222293778383568350588310650, 4.68967901835867317831478671084, 5.44113838224845248127704045827, 5.58888338268837600996760223866, 5.91943038126992156914162973044, 6.14640317875450093577799490037, 6.74687877622405378223541267975, 6.93524927378989757660266026624, 7.65494799383293181616186704631, 8.624863431129588132276124551128, 8.722047973217996408951500765928, 9.150034713544318670605475273109, 9.623661535681515484334411058219, 10.22055564903449068597804489576, 10.63855688308263044073699763356
