L(s) = 1 | + 2.73·3-s − 5-s − 3.39·7-s + 4.49·9-s + 2.78·11-s + 4.32·13-s − 2.73·15-s − 1.50·17-s − 3.42·19-s − 9.28·21-s + 3.48·23-s + 25-s + 4.08·27-s + 9.62·29-s − 5.68·31-s + 7.61·33-s + 3.39·35-s − 1.92·37-s + 11.8·39-s + 6.98·41-s − 2.45·43-s − 4.49·45-s − 5.87·47-s + 4.50·49-s − 4.11·51-s + 5.66·53-s − 2.78·55-s + ⋯ |
L(s) = 1 | + 1.58·3-s − 0.447·5-s − 1.28·7-s + 1.49·9-s + 0.838·11-s + 1.19·13-s − 0.706·15-s − 0.364·17-s − 0.786·19-s − 2.02·21-s + 0.726·23-s + 0.200·25-s + 0.786·27-s + 1.78·29-s − 1.02·31-s + 1.32·33-s + 0.573·35-s − 0.316·37-s + 1.89·39-s + 1.09·41-s − 0.374·43-s − 0.669·45-s − 0.857·47-s + 0.643·49-s − 0.576·51-s + 0.777·53-s − 0.375·55-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
\(L(1)\) |
\(\approx\) |
\(3.274295315\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.274295315\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 + 5.87T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 - 7.60T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.20T + 71T^{2} \) |
| 73 | \( 1 - 2.58T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 5.20T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.026127795687413003176300914419, −7.11480086682493041049423975817, −6.61866328784676669167833650337, −6.04910737257088363627566976612, −4.73654418104657615164855952225, −3.87790419654604447361229488113, −3.52412489256943359931320874516, −2.86807979862911048456588082131, −1.97515291917803273716833028935, −0.834558889788229773135878586604,
0.834558889788229773135878586604, 1.97515291917803273716833028935, 2.86807979862911048456588082131, 3.52412489256943359931320874516, 3.87790419654604447361229488113, 4.73654418104657615164855952225, 6.04910737257088363627566976612, 6.61866328784676669167833650337, 7.11480086682493041049423975817, 8.026127795687413003176300914419