L(s) = 1 | − 2.41·2-s − 3.18·3-s + 3.81·4-s + 5-s + 7.67·6-s − 3.73·7-s − 4.37·8-s + 7.11·9-s − 2.41·10-s + 11-s − 12.1·12-s + 1.00·13-s + 9.01·14-s − 3.18·15-s + 2.92·16-s + 3.53·17-s − 17.1·18-s + 5.00·19-s + 3.81·20-s + 11.8·21-s − 2.41·22-s − 0.391·23-s + 13.9·24-s + 25-s − 2.42·26-s − 13.1·27-s − 14.2·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 1.83·3-s + 1.90·4-s + 0.447·5-s + 3.13·6-s − 1.41·7-s − 1.54·8-s + 2.37·9-s − 0.762·10-s + 0.301·11-s − 3.50·12-s + 0.279·13-s + 2.40·14-s − 0.821·15-s + 0.730·16-s + 0.858·17-s − 4.04·18-s + 1.14·19-s + 0.853·20-s + 2.59·21-s − 0.514·22-s − 0.0816·23-s + 2.84·24-s + 0.200·25-s − 0.476·26-s − 2.52·27-s − 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4409068701\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4409068701\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 3.18T + 3T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 + 0.391T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 0.906T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 3.85T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 5.91T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 - 8.90T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 8.76T + 71T^{2} \) |
| 79 | \( 1 + 5.30T + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.597215922892708363721724532340, −7.51709125515896154373561810057, −7.00790317167696014115085384368, −6.27791199848613951268805191188, −5.95399076775963889947252045894, −5.04024774133025985607384368672, −3.78655911613187137098367425008, −2.58330331046380179498626359257, −1.16190240931778392829843440145, −0.66690340992398613091760487779,
0.66690340992398613091760487779, 1.16190240931778392829843440145, 2.58330331046380179498626359257, 3.78655911613187137098367425008, 5.04024774133025985607384368672, 5.95399076775963889947252045894, 6.27791199848613951268805191188, 7.00790317167696014115085384368, 7.51709125515896154373561810057, 8.597215922892708363721724532340