| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s + 7-s − 8-s + 2·10-s + 14-s + 2·15-s − 16-s + 21-s − 4·23-s − 24-s + 3·25-s − 27-s + 29-s + 2·30-s + 2·35-s − 2·40-s + 41-s + 42-s − 43-s − 4·46-s − 47-s − 48-s + 3·50-s − 54-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s + 7-s − 8-s + 2·10-s + 14-s + 2·15-s − 16-s + 21-s − 4·23-s − 24-s + 3·25-s − 27-s + 29-s + 2·30-s + 2·35-s − 2·40-s + 41-s + 42-s − 43-s − 4·46-s − 47-s − 48-s + 3·50-s − 54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.050445517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.050445517\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.835317581853808439331342035573, −9.672026816753842575800472634733, −9.469849714049458013692771237260, −8.850130508116658125473080670859, −8.329319116788229000019469189790, −8.177744549812294514381730914513, −7.994670558406957060091102247179, −7.03475843117515900663027734047, −6.69369900147518386166570067170, −5.95100985378178132397221195550, −5.92654807042718380568895107303, −5.63444315280561529586028488552, −4.97635837280244271141031581626, −4.38243749978159051448568449607, −4.32010509593046336158152976171, −3.39293151592411561808389669340, −3.07074017650230868218650159233, −2.22644591345794300815023612081, −2.14245237210292350252710280574, −1.52662382932611715115673485702,
1.52662382932611715115673485702, 2.14245237210292350252710280574, 2.22644591345794300815023612081, 3.07074017650230868218650159233, 3.39293151592411561808389669340, 4.32010509593046336158152976171, 4.38243749978159051448568449607, 4.97635837280244271141031581626, 5.63444315280561529586028488552, 5.92654807042718380568895107303, 5.95100985378178132397221195550, 6.69369900147518386166570067170, 7.03475843117515900663027734047, 7.994670558406957060091102247179, 8.177744549812294514381730914513, 8.329319116788229000019469189790, 8.850130508116658125473080670859, 9.469849714049458013692771237260, 9.672026816753842575800472634733, 9.835317581853808439331342035573
