L(s) = 1 | − 0.172·3-s − 5-s + 3.94·7-s − 2.97·9-s + 2.50·11-s + 5.08·13-s + 0.172·15-s − 5.79·17-s − 1.40·19-s − 0.680·21-s − 4.18·23-s + 25-s + 1.02·27-s + 3.24·29-s − 2.87·31-s − 0.431·33-s − 3.94·35-s − 2.92·37-s − 0.876·39-s − 11.1·41-s − 1.07·43-s + 2.97·45-s + 0.742·47-s + 8.59·49-s + 0.998·51-s − 7.39·53-s − 2.50·55-s + ⋯ |
L(s) = 1 | − 0.0995·3-s − 0.447·5-s + 1.49·7-s − 0.990·9-s + 0.755·11-s + 1.41·13-s + 0.0444·15-s − 1.40·17-s − 0.321·19-s − 0.148·21-s − 0.872·23-s + 0.200·25-s + 0.198·27-s + 0.603·29-s − 0.515·31-s − 0.0751·33-s − 0.667·35-s − 0.480·37-s − 0.140·39-s − 1.73·41-s − 0.163·43-s + 0.442·45-s + 0.108·47-s + 1.22·49-s + 0.139·51-s − 1.01·53-s − 0.337·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 0.172T + 3T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 + 1.40T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 - 0.742T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 0.250T + 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 3.29T + 97T^{2} \) |
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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
−7.76133803229693405928079354030, −6.52668355508829717703644540930, −6.32506384303791033185827172605, −5.29477223434642541265898653232, −4.64952973366867017585231766451, −3.98123086628997995419731619888, −3.21888571232863784999616935648, −2.03090558005110247079134137473, −1.38523354952156025633250165150, 0,
1.38523354952156025633250165150, 2.03090558005110247079134137473, 3.21888571232863784999616935648, 3.98123086628997995419731619888, 4.64952973366867017585231766451, 5.29477223434642541265898653232, 6.32506384303791033185827172605, 6.52668355508829717703644540930, 7.76133803229693405928079354030