| L(s) = 1 | + 3-s − 5-s + 9-s + 11-s − 2·13-s − 15-s + 17-s + 2·27-s − 2·29-s + 33-s − 2·39-s − 45-s + 47-s + 51-s − 55-s + 2·65-s + 4·71-s − 2·73-s + 79-s + 2·81-s + 4·83-s − 85-s − 2·87-s − 2·97-s + 99-s + 103-s + 109-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 9-s + 11-s − 2·13-s − 15-s + 17-s + 2·27-s − 2·29-s + 33-s − 2·39-s − 45-s + 47-s + 51-s − 55-s + 2·65-s + 4·71-s − 2·73-s + 79-s + 2·81-s + 4·83-s − 85-s − 2·87-s − 2·97-s + 99-s + 103-s + 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.163049765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163049765\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 - T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.23392878286525140248780318120, −9.729004334091381238895373971674, −9.714787144576502262769115616118, −9.054658721821552633669598858701, −8.936392079636107408664329479193, −8.181185703704613724578390242460, −7.84855435558716677520218045728, −7.60003251189679879365899329311, −7.16475291803559823994934002902, −6.86429486114435973651228308372, −6.27612004249513889200114167210, −5.61160677826465504215446346353, −4.95256425028965314608457326350, −4.75105702172542097917886065769, −4.02175718402967530696118647528, −3.59140465650062306201763060742, −3.37946706796594466844216085683, −2.39815850090055086382958484749, −2.14814985536626506751662612022, −1.05583541691977230752345981379,
1.05583541691977230752345981379, 2.14814985536626506751662612022, 2.39815850090055086382958484749, 3.37946706796594466844216085683, 3.59140465650062306201763060742, 4.02175718402967530696118647528, 4.75105702172542097917886065769, 4.95256425028965314608457326350, 5.61160677826465504215446346353, 6.27612004249513889200114167210, 6.86429486114435973651228308372, 7.16475291803559823994934002902, 7.60003251189679879365899329311, 7.84855435558716677520218045728, 8.181185703704613724578390242460, 8.936392079636107408664329479193, 9.054658721821552633669598858701, 9.714787144576502262769115616118, 9.729004334091381238895373971674, 10.23392878286525140248780318120
