| L(s) = 1 | − 2-s − 1.09·3-s + 4-s − 2.81·5-s + 1.09·6-s − 7-s − 8-s − 1.80·9-s + 2.81·10-s − 3.14·11-s − 1.09·12-s − 1.81·13-s + 14-s + 3.07·15-s + 16-s − 1.36·17-s + 1.80·18-s + 7.68·19-s − 2.81·20-s + 1.09·21-s + 3.14·22-s − 4.80·23-s + 1.09·24-s + 2.90·25-s + 1.81·26-s + 5.25·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.631·3-s + 0.5·4-s − 1.25·5-s + 0.446·6-s − 0.377·7-s − 0.353·8-s − 0.600·9-s + 0.888·10-s − 0.947·11-s − 0.315·12-s − 0.502·13-s + 0.267·14-s + 0.794·15-s + 0.250·16-s − 0.329·17-s + 0.424·18-s + 1.76·19-s − 0.628·20-s + 0.238·21-s + 0.669·22-s − 1.00·23-s + 0.223·24-s + 0.580·25-s + 0.355·26-s + 1.01·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 - 9.25T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 + 0.481T + 59T^{2} \) |
| 61 | \( 1 + 7.80T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 0.876T + 71T^{2} \) |
| 73 | \( 1 + 9.59T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 9.79T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.69592480916592275335195295104, −7.32532561477920973387126348679, −6.35214585400982699791136241691, −5.63611521173510531720782932732, −4.96839277455212250079791020143, −3.98018645116130754759093567813, −3.11966919234164162802489128846, −2.38722812533980924195561043289, −0.795219689778825426920295647805, 0,
0.795219689778825426920295647805, 2.38722812533980924195561043289, 3.11966919234164162802489128846, 3.98018645116130754759093567813, 4.96839277455212250079791020143, 5.63611521173510531720782932732, 6.35214585400982699791136241691, 7.32532561477920973387126348679, 7.69592480916592275335195295104
