L(s) = 1 | + (−1.69 + 0.373i)3-s + (−0.203 + 1.15i)5-s + (2.02 + 1.70i)7-s + (2.72 − 1.26i)9-s + (3.15 − 0.555i)11-s + (−1.12 + 1.33i)13-s + (−0.0867 − 2.02i)15-s − 3.16·17-s − 6.65i·19-s + (−4.05 − 2.12i)21-s + (−3.65 + 4.35i)23-s + (3.40 + 1.23i)25-s + (−4.13 + 3.15i)27-s + (3.84 + 4.58i)29-s + (1.52 + 4.18i)31-s + ⋯ |
L(s) = 1 | + (−0.976 + 0.215i)3-s + (−0.0910 + 0.516i)5-s + (0.764 + 0.644i)7-s + (0.907 − 0.420i)9-s + (0.950 − 0.167i)11-s + (−0.311 + 0.371i)13-s + (−0.0223 − 0.524i)15-s − 0.767·17-s − 1.52i·19-s + (−0.885 − 0.464i)21-s + (−0.762 + 0.908i)23-s + (0.681 + 0.247i)25-s + (−0.795 + 0.606i)27-s + (0.714 + 0.851i)29-s + (0.273 + 0.751i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.839220 + 0.746237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.839220 + 0.746237i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 - 0.373i)T \) |
| 7 | \( 1 + (-2.02 - 1.70i)T \) |
good | 5 | \( 1 + (0.203 - 1.15i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-3.15 + 0.555i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.12 - 1.33i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 6.65iT - 19T^{2} \) |
| 23 | \( 1 + (3.65 - 4.35i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.84 - 4.58i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.52 - 4.18i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.75 - 8.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.74 - 6.50i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.80 - 1.38i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (7.95 + 2.89i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (11.7 - 6.78i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.39 + 7.04i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.888 - 2.43i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.66 + 15.1i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.08 - 1.20i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.01 - 0.583i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.74 - 9.88i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (5.29 - 4.44i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + (0.0688 - 0.189i)T + (-74.3 - 62.3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00409765981239111366791649104, −9.614210705026546637518906384462, −9.068904127206953489168608155163, −7.88540454372170516517321959469, −6.73154449566900846623486800792, −6.33164807789700937872817115626, −5.00060796657120799878710786494, −4.47655502565090426828251524189, −2.98212358129779391864313372423, −1.42851273427336354772158242967,
0.72545918001127836238797095549, 1.97975906015787284317736904516, 4.15681747109737653095107456069, 4.51975108145811861831000324368, 5.77638452826596442753689338059, 6.50968854197363188147306657949, 7.59173802161385141636601768094, 8.223455772682795802609823181488, 9.425847798241634705755079327116, 10.33302539583987161953661035146