| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s + 11-s − 13-s − 16-s + 2·17-s + 22-s − 23-s − 26-s + 29-s + 31-s + 2·34-s − 4·37-s + 40-s − 43-s − 46-s − 47-s − 49-s − 55-s + 58-s − 2·59-s + 62-s + 64-s + 65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s + 11-s − 13-s − 16-s + 2·17-s + 22-s − 23-s − 26-s + 29-s + 31-s + 2·34-s − 4·37-s + 40-s − 43-s − 46-s − 47-s − 49-s − 55-s + 58-s − 2·59-s + 62-s + 64-s + 65-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.263087853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263087853\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.151249437841922681145230368014, −8.328841432101310499782599272380, −8.199630352896947043010679864993, −8.145273314305044598185963799598, −7.51884823665551572020554574559, −6.96776127452110724148730143219, −6.78620523954121927844545648154, −6.44525647106268202295484074576, −5.90736231249379593415394111778, −5.45326559416163228544252583361, −5.17807948522031802136723805143, −4.64231365639207788635566454689, −4.56164372473304840249343175666, −3.82692719228846909837839356870, −3.57046981248565124439813858206, −3.18297503559046341730402291264, −3.01539536230056019024095962902, −1.95712690178876785351690882242, −1.64134411004416809183873858310, −0.57968034872883906694187882052,
0.57968034872883906694187882052, 1.64134411004416809183873858310, 1.95712690178876785351690882242, 3.01539536230056019024095962902, 3.18297503559046341730402291264, 3.57046981248565124439813858206, 3.82692719228846909837839356870, 4.56164372473304840249343175666, 4.64231365639207788635566454689, 5.17807948522031802136723805143, 5.45326559416163228544252583361, 5.90736231249379593415394111778, 6.44525647106268202295484074576, 6.78620523954121927844545648154, 6.96776127452110724148730143219, 7.51884823665551572020554574559, 8.145273314305044598185963799598, 8.199630352896947043010679864993, 8.328841432101310499782599272380, 9.151249437841922681145230368014
