| L(s) = 1 | + 6.61i·5-s − 9.43·7-s + 17.6i·11-s − 20.6·13-s + 16.0i·17-s − 12.2·19-s − 26.6i·23-s − 18.6·25-s + 0.921i·29-s − 53.7·31-s − 62.3i·35-s + 64.0·37-s + 19.7i·41-s + 69.7·43-s − 0.606i·47-s + ⋯ |
| L(s) = 1 | + 1.32i·5-s − 1.34·7-s + 1.60i·11-s − 1.59·13-s + 0.944i·17-s − 0.647·19-s − 1.16i·23-s − 0.747·25-s + 0.0317i·29-s − 1.73·31-s − 1.78i·35-s + 1.73·37-s + 0.482i·41-s + 1.62·43-s − 0.0128i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07593904509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07593904509\) |
| \(L(2)\) |
|
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 | \( 1 - 6.61iT - 25T^{2} \) |
| 7 | \( 1 + 9.43T + 49T^{2} \) |
| 11 | \( 1 - 17.6iT - 121T^{2} \) |
| 13 | \( 1 + 20.6T + 169T^{2} \) |
| 17 | \( 1 - 16.0iT - 289T^{2} \) |
| 19 | \( 1 + 12.2T + 361T^{2} \) |
| 23 | \( 1 + 26.6iT - 529T^{2} \) |
| 29 | \( 1 - 0.921iT - 841T^{2} \) |
| 31 | \( 1 + 53.7T + 961T^{2} \) |
| 37 | \( 1 - 64.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 19.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 69.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 0.606iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 45.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 71.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 7.43T + 4.48e3T^{2} \) |
| 71 | \( 1 + 27.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 6.30T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 72.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 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
−9.660691870091225447552068854101, −9.353879005553768605949216394991, −7.86890085843079455747008074583, −7.18501571322327267629193699072, −6.66795024501656210084290495027, −5.98099966097950173224646861485, −4.66738380045032649810802480533, −3.83954995160745594989948862400, −2.69616659019086091228299412168, −2.20202348068984787004493138438,
0.02578625613428767159715936349, 0.809483728084877836965253625909, 2.44769000133787365613452589839, 3.39760704051869694628612351890, 4.37722581608409588976742891892, 5.42450604439048560769469490142, 5.86999939586893540167871499767, 7.02253964938795127939679708115, 7.75505756292425129467326502979, 8.765625661333344366928051689034
