L(s) = 1 | + 1.41·2-s + (−1.07 + 2.79i)3-s + 2.00·4-s + (2.52 + 4.31i)5-s + (−1.52 + 3.95i)6-s + 2.64i·7-s + 2.82·8-s + (−6.67 − 6.03i)9-s + (3.57 + 6.10i)10-s − 5.98i·11-s + (−2.15 + 5.59i)12-s + 19.0i·13-s + 3.74i·14-s + (−14.8 + 2.42i)15-s + 4.00·16-s − 2.02·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.359 + 0.933i)3-s + 0.500·4-s + (0.505 + 0.863i)5-s + (−0.253 + 0.659i)6-s + 0.377i·7-s + 0.353·8-s + (−0.742 − 0.670i)9-s + (0.357 + 0.610i)10-s − 0.544i·11-s + (−0.179 + 0.466i)12-s + 1.46i·13-s + 0.267i·14-s + (−0.986 + 0.161i)15-s + 0.250·16-s − 0.119·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36463 + 1.60615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36463 + 1.60615i\) |
\(L(2)\) |
|
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 | \( 1 + (1.07 - 2.79i)T \) |
| 5 | \( 1 + (-2.52 - 4.31i)T \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 + 5.98iT - 121T^{2} \) |
| 13 | \( 1 - 19.0iT - 169T^{2} \) |
| 17 | \( 1 + 2.02T + 289T^{2} \) |
| 19 | \( 1 + 21.1T + 361T^{2} \) |
| 23 | \( 1 - 32.2T + 529T^{2} \) |
| 29 | \( 1 + 12.2iT - 841T^{2} \) |
| 31 | \( 1 - 39.3T + 961T^{2} \) |
| 37 | \( 1 - 20.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 34.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 72.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 57.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 71.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 81.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 7.70iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 131.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 115.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 8.19iT - 9.40e3T^{2} \) |
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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
−12.24883747507897307376846713700, −11.28872483385332074045678697243, −10.74805303123273020791634913699, −9.632084451159593032240164291780, −8.664773308839641864334296451917, −6.80277980182915633035624635442, −6.14124212712715677621203440320, −4.95283891634250465676566074745, −3.77304993228262359500618924867, −2.46388764134611751521001552844,
1.05511584866339343269729563167, 2.64144147783494925432335122981, 4.59394176610587621352924198848, 5.54197602990407728891671691343, 6.54537621631395943501437928328, 7.66788147554426546268757167441, 8.661591088459332340166495976647, 10.17433194061905114283681454915, 11.08436500281358785751601905965, 12.30387423131418409550563786518