| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − 2.73i·7-s + i·8-s − 9-s + 0.267·11-s − i·12-s − 0.732i·13-s − 2.73·14-s + 16-s + 4.19i·17-s + i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.03i·7-s + 0.353i·8-s − 0.333·9-s + 0.0807·11-s − 0.288i·12-s − 0.203i·13-s − 0.730·14-s + 0.250·16-s + 1.01i·17-s + 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.625734700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625734700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 13 | \( 1 + 0.732iT - 13T^{2} \) |
| 17 | \( 1 - 4.19iT - 17T^{2} \) |
| 23 | \( 1 - 7.92iT - 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.19iT - 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 1.73iT - 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 + 3.73iT - 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 + 4.46iT - 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 0.267iT - 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 9.12iT - 97T^{2} \) |
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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
−9.004867101972416601421973233476, −7.952435955304618948835588696286, −7.47378592762178586248885896157, −6.31074215802531489618822421755, −5.49802935113631877818317993759, −4.58112555628060699119612630770, −3.82917997625488571070364715563, −3.29402079869067091185337139276, −1.98766430306102036390346906195, −0.828525079832537210064425432570,
0.76063953128108723011741886526, 2.25869142473229097200356071145, 3.00402867640629292525376570369, 4.38147611487047617980225135059, 5.08425831493630026748151401532, 6.01718902742684402155845326799, 6.44816063517950863936282896746, 7.36157546851479601911751716677, 7.978932857308810764246514162692, 8.833291584399714314947282616376
