| L(s) = 1 | − 2.56·3-s − 3.56·5-s + 2.56·7-s + 3.56·9-s − 2.56·11-s + 9.12·15-s − 5·17-s + 2.56·19-s − 6.56·21-s + 3.68·23-s + 7.68·25-s − 1.43·27-s − 5·29-s + 8·31-s + 6.56·33-s − 9.12·35-s − 37-s + 9.24·41-s + 6.56·43-s − 12.6·45-s − 4·47-s − 0.438·49-s + 12.8·51-s + 4.43·53-s + 9.12·55-s − 6.56·57-s + 2.56·59-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 1.59·5-s + 0.968·7-s + 1.18·9-s − 0.772·11-s + 2.35·15-s − 1.21·17-s + 0.587·19-s − 1.43·21-s + 0.768·23-s + 1.53·25-s − 0.276·27-s − 0.928·29-s + 1.43·31-s + 1.14·33-s − 1.54·35-s − 0.164·37-s + 1.44·41-s + 1.00·43-s − 1.89·45-s − 0.583·47-s − 0.0626·49-s + 1.79·51-s + 0.609·53-s + 1.23·55-s − 0.869·57-s + 0.333·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 9.24T + 41T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 + 7.24T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.249169078943114380213894314905, −7.61040276273750634236284507171, −7.03401257858780383813167782016, −6.10174803572499210922584177084, −5.14161687077441170328338407462, −4.67257958353095434834115931714, −3.97507229509595608839018308800, −2.65890713602005390669964368898, −1.04989136537845278309854557685, 0,
1.04989136537845278309854557685, 2.65890713602005390669964368898, 3.97507229509595608839018308800, 4.67257958353095434834115931714, 5.14161687077441170328338407462, 6.10174803572499210922584177084, 7.03401257858780383813167782016, 7.61040276273750634236284507171, 8.249169078943114380213894314905
