| L(s) = 1 | − 2.23·2-s + 2.94·3-s + 3.00·4-s − 1.14·5-s − 6.59·6-s − 0.894·7-s − 2.24·8-s + 5.69·9-s + 2.56·10-s + 1.44·11-s + 8.85·12-s − 2.17·13-s + 2.00·14-s − 3.38·15-s − 0.989·16-s − 5.56·17-s − 12.7·18-s − 7.39·19-s − 3.44·20-s − 2.63·21-s − 3.23·22-s − 2.96·23-s − 6.61·24-s − 3.67·25-s + 4.85·26-s + 7.93·27-s − 2.68·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.70·3-s + 1.50·4-s − 0.513·5-s − 2.69·6-s − 0.338·7-s − 0.792·8-s + 1.89·9-s + 0.812·10-s + 0.435·11-s + 2.55·12-s − 0.602·13-s + 0.534·14-s − 0.874·15-s − 0.247·16-s − 1.35·17-s − 3.00·18-s − 1.69·19-s − 0.771·20-s − 0.575·21-s − 0.689·22-s − 0.619·23-s − 1.34·24-s − 0.735·25-s + 0.952·26-s + 1.52·27-s − 0.507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1511 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 1.14T + 5T^{2} \) |
| 7 | \( 1 + 0.894T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 7.39T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 0.0594T + 31T^{2} \) |
| 37 | \( 1 - 0.869T + 37T^{2} \) |
| 41 | \( 1 + 0.928T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 - 5.79T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 2.85T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.921764832681490297940905624424, −8.459199691883784897794660920357, −7.84395520853923611024234797064, −7.09164517972394116206109222316, −6.41988829485587048821548904340, −4.44586690307947134903348789564, −3.73023275898194204143097320009, −2.41810077443060439291905678637, −1.88238630645213250158050257842, 0,
1.88238630645213250158050257842, 2.41810077443060439291905678637, 3.73023275898194204143097320009, 4.44586690307947134903348789564, 6.41988829485587048821548904340, 7.09164517972394116206109222316, 7.84395520853923611024234797064, 8.459199691883784897794660920357, 8.921764832681490297940905624424
