L(s) = 1 | + (2.75 − 1.18i)3-s + (−0.00985 − 0.00985i)5-s − 6.42i·7-s + (6.19 − 6.53i)9-s + (−9.07 − 9.07i)11-s + (−12.6 − 12.6i)13-s + (−0.0388 − 0.0154i)15-s + 19.0i·17-s + (−2.07 − 2.07i)19-s + (−7.61 − 17.7i)21-s + 19.5·23-s − 24.9i·25-s + (9.32 − 25.3i)27-s + (−11.1 + 11.1i)29-s + 59.9·31-s + ⋯ |
L(s) = 1 | + (0.918 − 0.395i)3-s + (−0.00197 − 0.00197i)5-s − 0.917i·7-s + (0.687 − 0.725i)9-s + (−0.824 − 0.824i)11-s + (−0.969 − 0.969i)13-s + (−0.00259 − 0.00103i)15-s + 1.11i·17-s + (−0.109 − 0.109i)19-s + (−0.362 − 0.842i)21-s + 0.850·23-s − 0.999i·25-s + (0.345 − 0.938i)27-s + (−0.385 + 0.385i)29-s + 1.93·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28897 - 1.48847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28897 - 1.48847i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.75 + 1.18i)T \) |
good | 5 | \( 1 + (0.00985 + 0.00985i)T + 25iT^{2} \) |
| 7 | \( 1 + 6.42iT - 49T^{2} \) |
| 11 | \( 1 + (9.07 + 9.07i)T + 121iT^{2} \) |
| 13 | \( 1 + (12.6 + 12.6i)T + 169iT^{2} \) |
| 17 | \( 1 - 19.0iT - 289T^{2} \) |
| 19 | \( 1 + (2.07 + 2.07i)T + 361iT^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 + (11.1 - 11.1i)T - 841iT^{2} \) |
| 31 | \( 1 - 59.9T + 961T^{2} \) |
| 37 | \( 1 + (9.32 - 9.32i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 47.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.1 + 24.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 6.29iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-20.6 - 20.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-60.3 - 60.3i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (48.0 + 48.0i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (23.7 + 23.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 13.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 31.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 47.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-70.3 + 70.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 95.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.6T + 9.40e3T^{2} \) |
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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
−10.37905351155046428425163144497, −10.31077353654848728896329729882, −8.793444960805550020753047330401, −8.060329802843874523894756118277, −7.33853162839051092560263184392, −6.24708125830073009232685932718, −4.83040574962255031753278637930, −3.54713963415670457698023952140, −2.53786703069061911042294985798, −0.77424029312596768867697179144,
2.11826264486098359505975224893, 2.93440654292767762805492180222, 4.54516094562743419090617900714, 5.22747099813702828032443339586, 6.91255828655572478303565465728, 7.67879044040349138839514097876, 8.780776507158175249284758575701, 9.505026472418997972582943599912, 10.13552476066955523282569113207, 11.43914001275211069454958458174