L(s) = 1 | + 2.49·2-s + 1.66·3-s + 4.22·4-s + 4.15·6-s − 4.24·7-s + 5.56·8-s − 0.229·9-s − 6.39·11-s + 7.03·12-s − 1.34·13-s − 10.6·14-s + 5.42·16-s + 1.79·17-s − 0.573·18-s − 3.80·19-s − 7.07·21-s − 15.9·22-s + 5.33·23-s + 9.25·24-s − 3.34·26-s − 5.37·27-s − 17.9·28-s + 1.62·29-s − 3.96·31-s + 2.41·32-s − 10.6·33-s + 4.48·34-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 0.960·3-s + 2.11·4-s + 1.69·6-s − 1.60·7-s + 1.96·8-s − 0.0766·9-s − 1.92·11-s + 2.03·12-s − 0.371·13-s − 2.83·14-s + 1.35·16-s + 0.436·17-s − 0.135·18-s − 0.872·19-s − 1.54·21-s − 3.40·22-s + 1.11·23-s + 1.88·24-s − 0.656·26-s − 1.03·27-s − 3.39·28-s + 0.301·29-s − 0.712·31-s + 0.426·32-s − 1.85·33-s + 0.769·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.66T + 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 2.25T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 4.44T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 + 9.81T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−7.30742529015895188816467106239, −7.04493314420361343796724067431, −6.05881281319895162451138988414, −5.47826204380946018494336199510, −4.87339097122353380553390160553, −3.78515843160698395804871406610, −3.24594255189450890586038361703, −2.71988423154295930923540106378, −2.17989385199709490022857553406, 0,
2.17989385199709490022857553406, 2.71988423154295930923540106378, 3.24594255189450890586038361703, 3.78515843160698395804871406610, 4.87339097122353380553390160553, 5.47826204380946018494336199510, 6.05881281319895162451138988414, 7.04493314420361343796724067431, 7.30742529015895188816467106239