L(s) = 1 | + 2-s + 3.45·3-s + 4-s + 5-s + 3.45·6-s + 0.963·7-s + 8-s + 8.92·9-s + 10-s + 11-s + 3.45·12-s − 4.86·13-s + 0.963·14-s + 3.45·15-s + 16-s + 5.97·17-s + 8.92·18-s − 6.74·19-s + 20-s + 3.32·21-s + 22-s + 1.31·23-s + 3.45·24-s + 25-s − 4.86·26-s + 20.4·27-s + 0.963·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.99·3-s + 0.5·4-s + 0.447·5-s + 1.40·6-s + 0.364·7-s + 0.353·8-s + 2.97·9-s + 0.316·10-s + 0.301·11-s + 0.996·12-s − 1.34·13-s + 0.257·14-s + 0.891·15-s + 0.250·16-s + 1.44·17-s + 2.10·18-s − 1.54·19-s + 0.223·20-s + 0.725·21-s + 0.213·22-s + 0.274·23-s + 0.704·24-s + 0.200·25-s − 0.954·26-s + 3.93·27-s + 0.182·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.565397961\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.565397961\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 - 3.45T + 3T^{2} \) |
| 7 | \( 1 - 0.963T + 7T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 6.76T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 - 0.511T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 + 1.81T + 83T^{2} \) |
| 89 | \( 1 + 2.02T + 89T^{2} \) |
| 97 | \( 1 + 4.71T + 97T^{2} \) |
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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
−7.87089567176463901395916370446, −7.23035319594581983239876613684, −6.67215875872203705959047840195, −5.64480337298558019656623226765, −4.71694388486797561818075204070, −4.22550630818619016706412054959, −3.40197367045652462425399332396, −2.69852454918161485033671230080, −2.11574953472606924772891345700, −1.34907088830840382777257672531,
1.34907088830840382777257672531, 2.11574953472606924772891345700, 2.69852454918161485033671230080, 3.40197367045652462425399332396, 4.22550630818619016706412054959, 4.71694388486797561818075204070, 5.64480337298558019656623226765, 6.67215875872203705959047840195, 7.23035319594581983239876613684, 7.87089567176463901395916370446