| L(s) = 1 | + 3.78i·5-s + 0.953·7-s + 11.6i·11-s + 8.69·13-s + 4.73i·17-s + 29.2·19-s − 2.69i·23-s + 10.6·25-s − 21.7i·29-s − 22.5·31-s + 3.60i·35-s − 24.0·37-s − 19.7i·41-s + 49.0·43-s + 59.3i·47-s + ⋯ |
| L(s) = 1 | + 0.756i·5-s + 0.136·7-s + 1.06i·11-s + 0.668·13-s + 0.278i·17-s + 1.54·19-s − 0.117i·23-s + 0.427·25-s − 0.748i·29-s − 0.728·31-s + 0.103i·35-s − 0.651·37-s − 0.482i·41-s + 1.13·43-s + 1.26i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.019559686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.019559686\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.78iT - 25T^{2} \) |
| 7 | \( 1 - 0.953T + 49T^{2} \) |
| 11 | \( 1 - 11.6iT - 121T^{2} \) |
| 13 | \( 1 - 8.69T + 169T^{2} \) |
| 17 | \( 1 - 4.73iT - 289T^{2} \) |
| 19 | \( 1 - 29.2T + 361T^{2} \) |
| 23 | \( 1 + 2.69iT - 529T^{2} \) |
| 29 | \( 1 + 21.7iT - 841T^{2} \) |
| 31 | \( 1 + 22.5T + 961T^{2} \) |
| 37 | \( 1 + 24.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 59.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 12.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 111.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 56.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 144.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 19.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 27T + 9.40e3T^{2} \) |
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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
−9.476069431002764004455653472036, −8.533028829908022706938332193006, −7.55294032352364868792609684134, −7.09476996274921682612143776862, −6.16173775478890327829366442188, −5.30962906932197300350749348809, −4.29679268845706661283330271590, −3.37035187518789712285035059017, −2.39631378264428614320485093558, −1.20866394708854336019634373176,
0.60412197377942042526976511238, 1.53386171488895436323060172123, 3.05869068039332385126096807816, 3.78266308402273078472094955735, 5.05807960176426955043987944897, 5.47798538143044567363920390888, 6.50712331723012103521946499382, 7.43517300935323258678427100428, 8.300017502609142871397880849163, 8.885456056307865949834754134449
