| L(s) = 1 | − 1.41·2-s + (−0.922 + 2.85i)3-s + 2.00·4-s + (1.30 − 4.03i)6-s − 2.64i·7-s − 2.82·8-s + (−7.29 − 5.26i)9-s − 4.58i·11-s + (−1.84 + 5.70i)12-s + 20.4i·13-s + 3.74i·14-s + 4.00·16-s + 4.97·17-s + (10.3 + 7.45i)18-s − 6.22·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.307 + 0.951i)3-s + 0.500·4-s + (0.217 − 0.672i)6-s − 0.377i·7-s − 0.353·8-s + (−0.810 − 0.585i)9-s − 0.416i·11-s + (−0.153 + 0.475i)12-s + 1.57i·13-s + 0.267i·14-s + 0.250·16-s + 0.292·17-s + (0.573 + 0.413i)18-s − 0.327·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4051930622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4051930622\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (0.922 - 2.85i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 + 4.58iT - 121T^{2} \) |
| 13 | \( 1 - 20.4iT - 169T^{2} \) |
| 17 | \( 1 - 4.97T + 289T^{2} \) |
| 19 | \( 1 + 6.22T + 361T^{2} \) |
| 23 | \( 1 + 0.140T + 529T^{2} \) |
| 29 | \( 1 + 1.76iT - 841T^{2} \) |
| 31 | \( 1 + 29.7T + 961T^{2} \) |
| 37 | \( 1 - 38.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 68.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 57.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 41.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 79.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 67.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 30.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 111. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 156.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 133. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.1iT - 9.40e3T^{2} \) |
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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
−9.472647432422030217827415716044, −8.959303680734591699885986570483, −8.090857124258827790766922838375, −6.95251310392311184695599413057, −6.25783900008738661611817503800, −5.18151723831329714169172580208, −4.17597555979730271856539216606, −3.29017139148426645984838023730, −1.80712930194226351341442426646, −0.18343486493551746588361052811,
1.09751184203027561594624667575, 2.27154633876006511086146748298, 3.26330386757553451647157287369, 5.02044260826615195704169110198, 5.84817478221612877889517339795, 6.61788974530747458655935078936, 7.67915298125468857166163389531, 8.002071000777938658965528579493, 8.981731320668352939385692759863, 9.896206118230154099653236739925
