| L(s) = 1 | + 3.65i·3-s + 9.09·7-s − 4.33·9-s + (9.88 − 4.82i)11-s − 22.5·13-s − 22.9·17-s + 16.0i·19-s + 33.1i·21-s + 0.141i·23-s + 17.0i·27-s + 29.8i·29-s + 21.5·31-s + (17.6 + 36.1i)33-s + 28.0i·37-s − 82.1i·39-s + ⋯ |
| L(s) = 1 | + 1.21i·3-s + 1.29·7-s − 0.481·9-s + (0.898 − 0.438i)11-s − 1.73·13-s − 1.34·17-s + 0.842i·19-s + 1.58i·21-s + 0.00616i·23-s + 0.631i·27-s + 1.02i·29-s + 0.694·31-s + (0.533 + 1.09i)33-s + 0.757i·37-s − 2.10i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.709374309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709374309\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-9.88 + 4.82i)T \) |
| good | 3 | \( 1 - 3.65iT - 9T^{2} \) |
| 7 | \( 1 - 9.09T + 49T^{2} \) |
| 13 | \( 1 + 22.5T + 169T^{2} \) |
| 17 | \( 1 + 22.9T + 289T^{2} \) |
| 19 | \( 1 - 16.0iT - 361T^{2} \) |
| 23 | \( 1 - 0.141iT - 529T^{2} \) |
| 29 | \( 1 - 29.8iT - 841T^{2} \) |
| 31 | \( 1 - 21.5T + 961T^{2} \) |
| 37 | \( 1 - 28.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 15.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 75.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 50.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 98.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 125. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 100.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 128.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 75.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 58.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 62.4iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00244001895594679156616257004, −9.221746047055255385664517753645, −8.542556610507186990483501924129, −7.60503019856064384041457730445, −6.64309419878968862908309975907, −5.40752220474926239146784827649, −4.61452585551185006452004372373, −4.18816644826489920344526431917, −2.83213763239398098199143639631, −1.50847510318990700885048853949,
0.51076837911972569403423972119, 1.88860762506096576122491284068, 2.38666061739476321998377517308, 4.31808119061153135997195090634, 4.87045522364846853892060041726, 6.15214271066761009264261221580, 7.14940968883902932305311846707, 7.37879746288412492991243245812, 8.396703568636507437192242396323, 9.170938679405235688417221296946
