L(s) = 1 | − 5·5-s − 24.4·7-s + 28.9·11-s − 65.3·13-s + 68.1·17-s − 104.·19-s + 154.·23-s + 25·25-s − 205.·29-s + 18.2·31-s + 122.·35-s − 337.·37-s − 195.·41-s − 334.·43-s + 5.00·47-s + 252.·49-s − 319.·53-s − 144.·55-s − 430.·59-s + 594.·61-s + 326.·65-s − 195.·67-s − 425.·71-s + 929.·73-s − 707.·77-s − 24.4·79-s + 545.·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.31·7-s + 0.794·11-s − 1.39·13-s + 0.972·17-s − 1.26·19-s + 1.40·23-s + 0.200·25-s − 1.31·29-s + 0.105·31-s + 0.589·35-s − 1.50·37-s − 0.746·41-s − 1.18·43-s + 0.0155·47-s + 0.735·49-s − 0.829·53-s − 0.355·55-s − 0.950·59-s + 1.24·61-s + 0.623·65-s − 0.357·67-s − 0.711·71-s + 1.48·73-s − 1.04·77-s − 0.0347·79-s + 0.721·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8409352236\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8409352236\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 7 | \( 1 + 24.4T + 343T^{2} \) |
| 11 | \( 1 - 28.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 334.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 5.00T + 1.03e5T^{2} \) |
| 53 | \( 1 + 319.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 430.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 594.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 195.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 929.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 84.1T + 7.04e5T^{2} \) |
| 97 | \( 1 - 827.T + 9.12e5T^{2} \) |
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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
−8.863019401797787895741063399410, −7.892774677245693912004497915990, −6.94393319936921917783122333924, −6.67132495402933254129537036523, −5.53699321859943522663305933679, −4.69416957685383916777915939491, −3.61944691246452682243252028938, −3.08150735387415345350785728250, −1.83643310731236234038105935501, −0.40034842746650343121253446444,
0.40034842746650343121253446444, 1.83643310731236234038105935501, 3.08150735387415345350785728250, 3.61944691246452682243252028938, 4.69416957685383916777915939491, 5.53699321859943522663305933679, 6.67132495402933254129537036523, 6.94393319936921917783122333924, 7.892774677245693912004497915990, 8.863019401797787895741063399410