| L(s) = 1 | + (1.89 + 0.635i)2-s + (3.19 + 2.40i)4-s + 10.1i·7-s + (4.52 + 6.59i)8-s + 10.6i·11-s − 11.7·13-s + (−6.41 + 19.1i)14-s + (4.38 + 15.3i)16-s − 16.6·17-s − 0.464i·19-s + (−6.77 + 20.2i)22-s − 42.1i·23-s + (−22.3 − 7.47i)26-s + (−24.3 + 32.2i)28-s − 19.5·29-s + ⋯ |
| L(s) = 1 | + (0.948 + 0.317i)2-s + (0.798 + 0.602i)4-s + 1.44i·7-s + (0.565 + 0.824i)8-s + 0.968i·11-s − 0.905·13-s + (−0.458 + 1.36i)14-s + (0.274 + 0.961i)16-s − 0.978·17-s − 0.0244i·19-s + (−0.307 + 0.918i)22-s − 1.83i·23-s + (−0.858 − 0.287i)26-s + (−0.869 + 1.15i)28-s − 0.674·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.674029797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.674029797\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.89 - 0.635i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10.1iT - 49T^{2} \) |
| 11 | \( 1 - 10.6iT - 121T^{2} \) |
| 13 | \( 1 + 11.7T + 169T^{2} \) |
| 17 | \( 1 + 16.6T + 289T^{2} \) |
| 19 | \( 1 + 0.464iT - 361T^{2} \) |
| 23 | \( 1 + 42.1iT - 529T^{2} \) |
| 29 | \( 1 + 19.5T + 841T^{2} \) |
| 31 | \( 1 + 9.17iT - 961T^{2} \) |
| 37 | \( 1 + 23.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 58.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 63.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 22.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 61.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 15T + 9.40e3T^{2} \) |
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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
−10.34605187989717289655105110565, −9.268701860155184754668603405727, −8.517081294007924375577655680312, −7.50963878964409624275129573217, −6.66747840401709326958085795843, −5.85068600078851821902886364050, −4.91607544163532716520984275948, −4.25311898224451134028721600875, −2.64017195533188533425716002829, −2.20172956005156837303514096819,
0.57864695801608518556810805214, 1.97426817794889462005536959310, 3.37369529454665674088999762339, 4.02162052150745129389383227178, 5.05983703700884263249477817555, 5.92951794395610813014163165852, 7.13428259206593222829946977046, 7.39593015681911268728331176372, 8.824516282178154760161023492150, 9.928729284766025229481712091818
