| L(s) = 1 | − 3-s − i·5-s − 2i·7-s + 9-s + (−3 + 2i)13-s + i·15-s + 2·17-s + 2i·19-s + 2i·21-s + 8·23-s − 25-s − 27-s + 2·29-s − 2i·31-s − 2·35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447i·5-s − 0.755i·7-s + 0.333·9-s + (−0.832 + 0.554i)13-s + 0.258i·15-s + 0.485·17-s + 0.458i·19-s + 0.436i·21-s + 1.66·23-s − 0.200·25-s − 0.192·27-s + 0.371·29-s − 0.359i·31-s − 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372805289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372805289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.522190643727984003313862722355, −7.74573699960982392512849255391, −7.01428755976296296935111087025, −6.42989897170049001625091414730, −5.34286834548264854691785319378, −4.80460392965151878384217065687, −4.02069104658125692833253225044, −2.97923662493450803662057068877, −1.64308901727887977867242167015, −0.61843006633154194470074329297,
0.905879526300278179971308098793, 2.38568739790310813976339780757, 3.03727913813717020093379914053, 4.21269187224587705983816331964, 5.25312697425714408121065278330, 5.56227686197735496707960254253, 6.61732797681190909777125827112, 7.21084162857080359740598507710, 7.939369705998557132114838535044, 8.969616901324520201629794044262
