L(s) = 1 | + 2-s + 0.765·3-s + 4-s − 2.45·5-s + 0.765·6-s − 1.60·7-s + 8-s − 2.41·9-s − 2.45·10-s + 2.93·11-s + 0.765·12-s − 1.02·13-s − 1.60·14-s − 1.88·15-s + 16-s − 6.55·17-s − 2.41·18-s − 19-s − 2.45·20-s − 1.22·21-s + 2.93·22-s + 1.83·23-s + 0.765·24-s + 1.03·25-s − 1.02·26-s − 4.14·27-s − 1.60·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.442·3-s + 0.5·4-s − 1.09·5-s + 0.312·6-s − 0.606·7-s + 0.353·8-s − 0.804·9-s − 0.777·10-s + 0.884·11-s + 0.221·12-s − 0.283·13-s − 0.428·14-s − 0.485·15-s + 0.250·16-s − 1.59·17-s − 0.568·18-s − 0.229·19-s − 0.549·20-s − 0.267·21-s + 0.625·22-s + 0.381·23-s + 0.156·24-s + 0.207·25-s − 0.200·26-s − 0.797·27-s − 0.303·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.075753804\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.075753804\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 0.765T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 + 6.55T + 17T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 8.53T + 31T^{2} \) |
| 37 | \( 1 + 0.633T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 - 9.34T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 0.726T + 71T^{2} \) |
| 73 | \( 1 - 4.64T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.73278947443481924533439191246, −7.06399879934915964393211003698, −6.49862835296991326541248273661, −5.79545483737244033807536417203, −4.86153032027502097166212087357, −4.02333953179271020937030026299, −3.74040668017228409080810569285, −2.78828585462470986313198175650, −2.15963096345019677579143647793, −0.59372037997288478952457209548,
0.59372037997288478952457209548, 2.15963096345019677579143647793, 2.78828585462470986313198175650, 3.74040668017228409080810569285, 4.02333953179271020937030026299, 4.86153032027502097166212087357, 5.79545483737244033807536417203, 6.49862835296991326541248273661, 7.06399879934915964393211003698, 7.73278947443481924533439191246