| L(s) = 1 | + 2-s + 4-s − 0.350·5-s + 3.21·7-s + 8-s − 0.350·10-s − 3.33·11-s − 4.79·13-s + 3.21·14-s + 16-s + 1.65·17-s − 6.44·19-s − 0.350·20-s − 3.33·22-s − 7.13·23-s − 4.87·25-s − 4.79·26-s + 3.21·28-s − 2.80·29-s + 6.66·31-s + 32-s + 1.65·34-s − 1.12·35-s − 0.669·37-s − 6.44·38-s − 0.350·40-s − 0.589·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.156·5-s + 1.21·7-s + 0.353·8-s − 0.110·10-s − 1.00·11-s − 1.32·13-s + 0.859·14-s + 0.250·16-s + 0.402·17-s − 1.47·19-s − 0.0783·20-s − 0.710·22-s − 1.48·23-s − 0.975·25-s − 0.940·26-s + 0.607·28-s − 0.521·29-s + 1.19·31-s + 0.176·32-s + 0.284·34-s − 0.190·35-s − 0.110·37-s − 1.04·38-s − 0.0553·40-s − 0.0921·41-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 | \( 1 + 0.350T + 5T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + 0.669T + 37T^{2} \) |
| 41 | \( 1 + 0.589T + 41T^{2} \) |
| 43 | \( 1 + 4.08T + 43T^{2} \) |
| 47 | \( 1 + 7.50T + 47T^{2} \) |
| 53 | \( 1 + 7.08T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 - 9.99T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 6.63T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 8.59T + 83T^{2} \) |
| 89 | \( 1 + 0.107T + 89T^{2} \) |
| 97 | \( 1 + 0.0990T + 97T^{2} \) |
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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
−7.928153238813541454640333243234, −7.54792476369573929431789076853, −6.46815997173119057506594553197, −5.74298438202930424299759740171, −4.82013359082339417512264703863, −4.58056963094638260636840969690, −3.51034644409883099468811528026, −2.35370718607057698071628219632, −1.85624589239993067755664404811, 0,
1.85624589239993067755664404811, 2.35370718607057698071628219632, 3.51034644409883099468811528026, 4.58056963094638260636840969690, 4.82013359082339417512264703863, 5.74298438202930424299759740171, 6.46815997173119057506594553197, 7.54792476369573929431789076853, 7.928153238813541454640333243234
