| L(s) = 1 | − 2-s + 4-s + 2.93·5-s + 2.50·7-s − 8-s − 2.93·10-s − 4.59·11-s − 4.69·13-s − 2.50·14-s + 16-s + 1.37·17-s − 0.873·19-s + 2.93·20-s + 4.59·22-s − 3.75·23-s + 3.62·25-s + 4.69·26-s + 2.50·28-s − 5.21·29-s + 0.980·31-s − 32-s − 1.37·34-s + 7.36·35-s − 3.23·37-s + 0.873·38-s − 2.93·40-s − 5.95·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.31·5-s + 0.947·7-s − 0.353·8-s − 0.928·10-s − 1.38·11-s − 1.30·13-s − 0.669·14-s + 0.250·16-s + 0.334·17-s − 0.200·19-s + 0.656·20-s + 0.978·22-s − 0.781·23-s + 0.725·25-s + 0.920·26-s + 0.473·28-s − 0.968·29-s + 0.176·31-s − 0.176·32-s − 0.236·34-s + 1.24·35-s − 0.532·37-s + 0.141·38-s − 0.464·40-s − 0.929·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 0.873T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 0.980T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 5.45T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 3.87T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 - 5.16T + 97T^{2} \) |
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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
−7.973622273100387370568870107573, −7.60771203227378113768192768968, −6.72593234442903408503098864232, −5.70036811604819580974997186431, −5.33060590648866403708299548995, −4.50289996410224246416581467565, −2.99319709785220111912756857879, −2.18659447899924200275657394634, −1.63597781482082914912235561339, 0,
1.63597781482082914912235561339, 2.18659447899924200275657394634, 2.99319709785220111912756857879, 4.50289996410224246416581467565, 5.33060590648866403708299548995, 5.70036811604819580974997186431, 6.72593234442903408503098864232, 7.60771203227378113768192768968, 7.973622273100387370568870107573
