| L(s) = 1 | + 1.24·3-s − 3.80·5-s − 3.24·7-s − 1.44·9-s + 1.15·11-s − 6.04·13-s − 4.74·15-s − 4·17-s − 6.35·19-s − 4.04·21-s − 4.19·23-s + 9.45·25-s − 5.54·27-s + 6.29·31-s + 1.44·33-s + 12.3·35-s − 4.04·37-s − 7.54·39-s − 6.89·41-s − 0.417·43-s + 5.49·45-s − 4.76·47-s + 3.54·49-s − 4.98·51-s + 2.29·53-s − 4.40·55-s − 7.92·57-s + ⋯ |
| L(s) = 1 | + 0.719·3-s − 1.70·5-s − 1.22·7-s − 0.481·9-s + 0.349·11-s − 1.67·13-s − 1.22·15-s − 0.970·17-s − 1.45·19-s − 0.883·21-s − 0.875·23-s + 1.89·25-s − 1.06·27-s + 1.13·31-s + 0.251·33-s + 2.08·35-s − 0.665·37-s − 1.20·39-s − 1.07·41-s − 0.0637·43-s + 0.818·45-s − 0.694·47-s + 0.506·49-s − 0.698·51-s + 0.315·53-s − 0.594·55-s − 1.04·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 41 | \( 1 + 6.89T + 41T^{2} \) |
| 43 | \( 1 + 0.417T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 2.29T + 53T^{2} \) |
| 59 | \( 1 + 0.396T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4.78T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 5.85T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.35465706500007594573665806785, −6.72845028099445286432030051753, −6.11621435178453164351352408803, −4.79090849371993058245163763195, −4.28157520838625551222435173561, −3.50559696103168466396795923389, −2.90581331736289101487560639431, −2.12913686891503672013506480208, 0, 0,
2.12913686891503672013506480208, 2.90581331736289101487560639431, 3.50559696103168466396795923389, 4.28157520838625551222435173561, 4.79090849371993058245163763195, 6.11621435178453164351352408803, 6.72845028099445286432030051753, 7.35465706500007594573665806785
