| L(s) = 1 | − 2-s + (−1 − 1.41i)3-s + 4-s + i·5-s + (1 + 1.41i)6-s + 3.41·7-s − 8-s + (−1.00 + 2.82i)9-s − i·10-s − 2.58i·11-s + (−1 − 1.41i)12-s + 6.24i·13-s − 3.41·14-s + (1.41 − i)15-s + 16-s + 2.82i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.577 − 0.816i)3-s + 0.5·4-s + 0.447i·5-s + (0.408 + 0.577i)6-s + 1.29·7-s − 0.353·8-s + (−0.333 + 0.942i)9-s − 0.316i·10-s − 0.779i·11-s + (−0.288 − 0.408i)12-s + 1.73i·13-s − 0.912·14-s + (0.365 − 0.258i)15-s + 0.250·16-s + 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.903618 + 0.175659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.903618 + 0.175659i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (4.24 + i)T \) |
| good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.58iT - 11T^{2} \) |
| 13 | \( 1 - 6.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 9.07T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 + 8.82iT - 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 9.31iT - 79T^{2} \) |
| 83 | \( 1 - 7.17iT - 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 9.41iT - 97T^{2} \) |
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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
−10.99044168036423299835057418374, −10.10605250084482035912886282758, −8.658180644461699747566894654653, −8.277263154047694248599380089730, −7.14946053955124206373481033098, −6.52713972028414602226997519436, −5.50175612034016594296986682554, −4.22122000047918702823790887336, −2.33839329309589272615354702397, −1.35657485493794321942914727266,
0.804001058421859420290469090962, 2.58225121220856977310922473630, 4.30541185027600792492673443088, 5.03887163466081568366153761917, 5.98540694012270788031111082900, 7.27667189790026424773862825380, 8.297655785930812812975445660672, 8.823480936306068995985327888705, 10.10049205314492014644531484981, 10.46806142761083857396304238053
