| L(s) = 1 | + 2-s + 4-s + (1.35 − 2.27i)7-s + 8-s − 2.71i·11-s − 6.54·13-s + (1.35 − 2.27i)14-s + 16-s − 1.53i·17-s − 2.30i·19-s − 2.71i·22-s − 3.83·23-s − 6.54·26-s + (1.35 − 2.27i)28-s − 3.83i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.512 − 0.858i)7-s + 0.353·8-s − 0.817i·11-s − 1.81·13-s + (0.362 − 0.607i)14-s + 0.250·16-s − 0.371i·17-s − 0.528i·19-s − 0.577i·22-s − 0.799·23-s − 1.28·26-s + (0.256 − 0.429i)28-s − 0.711i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.806950293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.806950293\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.35 + 2.27i)T \) |
| good | 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 + 6.54T + 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 + 2.30iT - 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 + 3.83iT - 29T^{2} \) |
| 31 | \( 1 - 3.25iT - 31T^{2} \) |
| 37 | \( 1 + 3.01iT - 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 0.468iT - 43T^{2} \) |
| 47 | \( 1 - 9.11iT - 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 4.78iT - 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 + 2.30iT - 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.053350049993478436840076867637, −7.69015754326173372395675709047, −6.86032790742825215108862003154, −6.15537213183624709438406280426, −5.02077230288607430970458112757, −4.74035107979608636157105182828, −3.71192994878061393164178472531, −2.83566115920167901717179092366, −1.84498342170850924875159663402, −0.38656616488429029136967595805,
1.79604950851459884046234479822, 2.35718366561643472085788156160, 3.41567140822493520066403196785, 4.54945455085732201141903341514, 4.99148618556121377600836081687, 5.76920119003353640781825785047, 6.62815907088103620692838855958, 7.45686541821454645138046800426, 8.011380591241677664234627700373, 8.927215016903822288478334608998
