L(s) = 1 | + i·2-s + (1.53 − 0.801i)3-s − 4-s + 5-s + (0.801 + 1.53i)6-s − i·8-s + (1.71 − 2.46i)9-s + i·10-s + 5.48i·11-s + (−1.53 + 0.801i)12-s + 4.37i·13-s + (1.53 − 0.801i)15-s + 16-s − 6.99·17-s + (2.46 + 1.71i)18-s + 4.64i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.886 − 0.462i)3-s − 0.5·4-s + 0.447·5-s + (0.327 + 0.626i)6-s − 0.353i·8-s + (0.571 − 0.820i)9-s + 0.316i·10-s + 1.65i·11-s + (−0.443 + 0.231i)12-s + 1.21i·13-s + (0.396 − 0.206i)15-s + 0.250·16-s − 1.69·17-s + (0.580 + 0.404i)18-s + 1.06i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0595 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0595 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.148264640\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148264640\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.53 + 0.801i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.48iT - 11T^{2} \) |
| 13 | \( 1 - 4.37iT - 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 19 | \( 1 - 4.64iT - 19T^{2} \) |
| 23 | \( 1 - 7.99iT - 23T^{2} \) |
| 29 | \( 1 + 3.40iT - 29T^{2} \) |
| 31 | \( 1 + 3.85iT - 31T^{2} \) |
| 37 | \( 1 - 9.25T + 37T^{2} \) |
| 41 | \( 1 - 4.36T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 - 7.51T + 47T^{2} \) |
| 53 | \( 1 - 4.68iT - 53T^{2} \) |
| 59 | \( 1 - 6.00T + 59T^{2} \) |
| 61 | \( 1 + 6.75iT - 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 - 4.42iT - 71T^{2} \) |
| 73 | \( 1 + 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 + 9.22T + 83T^{2} \) |
| 89 | \( 1 - 0.525T + 89T^{2} \) |
| 97 | \( 1 + 2.60iT - 97T^{2} \) |
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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
−9.412192726566890319990085342312, −9.056018731365537684009890362912, −7.899414342983958987839710605974, −7.34000382765846318673982251267, −6.64188095259051111544403891797, −5.86756468359003077826778109884, −4.46188324127981005005991485452, −4.03952224889516527981499961281, −2.38824110342141800498314224383, −1.66808566339502291478152375419,
0.77269980381626976427152649982, 2.51561231604682575882969705502, 2.86179318235264413439659400827, 4.03241290222529713338877252498, 4.87409574626638398736598530349, 5.85832517628970742836149987912, 6.90017313197991719605495560367, 8.143917112006342602542648108611, 8.725677067879512040890674577102, 9.133035471954188492552589357866