| L(s) = 1 | + 1.56·3-s + (−2 − i)5-s + (2 − 1.73i)7-s − 0.561·9-s + 6.16·11-s − 3.56i·13-s + (−3.12 − 1.56i)15-s − 2.70·17-s + 7.12i·19-s + (3.12 − 2.70i)21-s + 7.12·23-s + (3 + 4i)25-s − 5.56·27-s − 2.70i·29-s + 5.40·31-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + (−0.894 − 0.447i)5-s + (0.755 − 0.654i)7-s − 0.187·9-s + 1.85·11-s − 0.987i·13-s + (−0.806 − 0.403i)15-s − 0.655·17-s + 1.63i·19-s + (0.681 − 0.590i)21-s + 1.48·23-s + (0.600 + 0.800i)25-s − 1.07·27-s − 0.502i·29-s + 0.971·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.355852205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.355852205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 + 3.56iT - 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 - 7.12iT - 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 2.70iT - 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 + 5.40iT - 41T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 + 6.16iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 2.24iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 4.68iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 5.40iT - 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.641500761240700078194633642198, −8.350156452006213568796873976081, −7.50492304756792591767272338984, −6.85343325517280472599693166027, −5.68150773444847940738289131378, −4.67567226295917019372382187684, −3.78219246710198752725632890346, −3.42652389102651499912680496947, −1.90104758115410901003272714907, −0.821855632193230530375970708790,
1.33197582917163382497346347933, 2.56419545049002654765519262348, 3.25703917928437174134049074024, 4.33908001293894669953438858107, 4.82329716037106567272512506753, 6.41345940834730954386675104829, 6.80661949469646646600187100520, 7.72246687516331026103701952416, 8.618585236501486904419779943642, 9.003023146230847039226651893557
