| L(s) = 1 | + (−250 − 125i)5-s + 722i·7-s + 3.99e3·11-s + 3.03e3i·13-s − 2.05e4i·17-s + 2.53e4·19-s − 6.66e4i·23-s + (4.68e4 + 6.25e4i)25-s − 1.52e5·29-s − 1.23e5·31-s + (9.02e4 − 1.80e5i)35-s + 3.37e5i·37-s − 3.96e5·41-s + 4.42e5i·43-s + 1.70e5i·47-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s + 0.795i·7-s + 0.904·11-s + 0.382i·13-s − 1.01i·17-s + 0.846·19-s − 1.14i·23-s + (0.599 + 0.799i)25-s − 1.16·29-s − 0.746·31-s + (0.355 − 0.711i)35-s + 1.09i·37-s − 0.898·41-s + 0.849i·43-s + 0.239i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3878441947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3878441947\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (250 + 125i)T \) |
| good | 7 | \( 1 - 722iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 3.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.03e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.05e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.53e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.66e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.23e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.37e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.42e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.70e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.23e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 3.02e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.83e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.74e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.00e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.40e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 7.51e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.29e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 7.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.00e7iT - 8.07e13T^{2} \) |
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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
−11.86171847026634351430671352350, −11.08971006696745304097515581547, −9.482514805404702157579857426051, −8.876677708870349062858585918377, −7.74516178536508037646133817443, −6.67024807192585293947618394415, −5.30597810307436906077004231667, −4.23238173185711688885304875640, −2.95648999846202006615127155673, −1.32697229504227886533526677975,
0.10616576183936039173569354817, 1.51070596144674787115739690215, 3.43075825681166467753890605453, 4.04087301358319925743917687115, 5.64411174901586297619185261546, 7.02682034204539483424304163170, 7.64166925900676663574935570567, 8.866601705884630720654249394431, 10.07511645293896848159055621527, 11.02980655422956737428135054231
