| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s − 16-s + 2·19-s + 23-s + 3·31-s + 2·38-s + 40-s + 46-s + 2·47-s + 49-s + 2·53-s − 3·61-s + 3·62-s + 64-s + 3·79-s + 80-s − 3·83-s + 2·94-s − 2·95-s + 98-s + 2·106-s − 115-s + 121-s − 3·122-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s − 16-s + 2·19-s + 23-s + 3·31-s + 2·38-s + 40-s + 46-s + 2·47-s + 49-s + 2·53-s − 3·61-s + 3·62-s + 64-s + 3·79-s + 80-s − 3·83-s + 2·94-s − 2·95-s + 98-s + 2·106-s − 115-s + 121-s − 3·122-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.829421624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.829421624\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−8.872271334701190448041472904085, −8.811558973847702002655833772939, −8.186860105875462530177872551970, −7.83742917416974922249155223262, −7.50192707701956675179681701891, −7.21109525321414142132101978870, −6.72186047110978722716552833667, −6.37538762640902366912605595907, −5.78625466329583375377506267025, −5.63689739952118966535027718403, −5.12441084845364472506472216577, −4.68556014356963110531476053953, −4.44139765229329629851185928633, −3.99993929063249794312464487497, −3.55801591570220950716833100703, −3.17829903633742274743160476213, −2.65501909914820034429238370499, −2.48135939208053609214986728750, −1.21172007572193496484057924199, −0.827728302549680570199567503576,
0.827728302549680570199567503576, 1.21172007572193496484057924199, 2.48135939208053609214986728750, 2.65501909914820034429238370499, 3.17829903633742274743160476213, 3.55801591570220950716833100703, 3.99993929063249794312464487497, 4.44139765229329629851185928633, 4.68556014356963110531476053953, 5.12441084845364472506472216577, 5.63689739952118966535027718403, 5.78625466329583375377506267025, 6.37538762640902366912605595907, 6.72186047110978722716552833667, 7.21109525321414142132101978870, 7.50192707701956675179681701891, 7.83742917416974922249155223262, 8.186860105875462530177872551970, 8.811558973847702002655833772939, 8.872271334701190448041472904085
