| L(s) = 1 | − 1.73i·3-s + 2·4-s + (−2.5 − 0.866i)7-s − 2.99·9-s − 3.46i·12-s − 5.19i·13-s + 4·16-s − 8.66i·19-s + (−1.49 + 4.33i)21-s + 5.19i·27-s + (−5 − 1.73i)28-s + 8.66i·31-s − 5.99·36-s + 10·37-s − 9·39-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 4-s + (−0.944 − 0.327i)7-s − 0.999·9-s − 0.999i·12-s − 1.44i·13-s + 16-s − 1.98i·19-s + (−0.327 + 0.944i)21-s + 0.999i·27-s + (−0.944 − 0.327i)28-s + 1.55i·31-s − 0.999·36-s + 1.64·37-s − 1.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.850514 - 1.19472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850514 - 1.19472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8.66iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 19.0iT - 97T^{2} \) |
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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
−10.80147448811252278177687358691, −9.823174331971979521554383397972, −8.621549860307797392169778950770, −7.64363926778765403064156486729, −6.91156973580227517567359219569, −6.28694311484491361433990075268, −5.23097121185261988116815359526, −3.24023173503805799586740935190, −2.57306711603478942721687942727, −0.836594478994325832410460183014,
2.13682601168984960613523377315, 3.35252138520686902433370894219, 4.26828206965667242726157074155, 5.85015050377251278015152898032, 6.25688266514178799696251230471, 7.50316042614573530045073321190, 8.589020569303564902161163192085, 9.692436918001818396013384138378, 10.03660340024899186878097966415, 11.18506512425431643785038834761
