| L(s) = 1 | + 2-s + 4-s + (−2.23 − 1.41i)7-s + 8-s + 1.41i·11-s − 5.39·13-s + (−2.23 − 1.41i)14-s + 16-s − 2.23i·17-s − 1.30i·19-s + 1.41i·22-s + 23-s − 5.39·26-s + (−2.23 − 1.41i)28-s + 9.24i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.845 − 0.534i)7-s + 0.353·8-s + 0.426i·11-s − 1.49·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s − 0.300i·19-s + 0.301i·22-s + 0.208·23-s − 1.05·26-s + (−0.422 − 0.267i)28-s + 1.71i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.104797297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104797297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 + 1.30iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 9.24iT - 29T^{2} \) |
| 31 | \( 1 - 8.56iT - 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 - 6.41iT - 43T^{2} \) |
| 47 | \( 1 - 7.63iT - 47T^{2} \) |
| 53 | \( 1 + 6.07T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.39iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4.34iT - 71T^{2} \) |
| 73 | \( 1 + 5.01T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 + 8.01iT - 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
| 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
−9.166611544056358035720715383692, −7.934531822071082413479009251490, −7.08406993035804116759588512828, −6.89625873310842114775374219359, −5.85755055993511969299193099833, −4.85309088737823201230750857112, −4.51497649071041394476138308521, −3.19299183418908177482572634952, −2.80454516223016457762985062587, −1.40505562246421168005337828059,
0.24824282207563417968294495053, 2.10180467581432643270710542935, 2.73967783969234922599297110963, 3.73560250205082956434627126805, 4.48512242824161871768873289796, 5.53284874899074038193057976610, 5.98199176503091750432318755332, 6.79797531642429498734267134810, 7.60358376042590070124177858308, 8.294791306159265906749238119579
