| L(s) = 1 | + 0.702i·2-s − 1.73·3-s + 3.50·4-s + (−0.979 − 4.90i)5-s − 1.21i·6-s + (3.48 − 6.07i)7-s + 5.27i·8-s + 2.99·9-s + (3.44 − 0.687i)10-s + 5.39·11-s − 6.07·12-s + 12.5·13-s + (4.26 + 2.44i)14-s + (1.69 + 8.49i)15-s + 10.3·16-s − 8.14·17-s + ⋯ |
| L(s) = 1 | + 0.351i·2-s − 0.577·3-s + 0.876·4-s + (−0.195 − 0.980i)5-s − 0.202i·6-s + (0.497 − 0.867i)7-s + 0.659i·8-s + 0.333·9-s + (0.344 − 0.0687i)10-s + 0.490·11-s − 0.506·12-s + 0.963·13-s + (0.304 + 0.174i)14-s + (0.113 + 0.566i)15-s + 0.645·16-s − 0.479·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.39417 - 0.227254i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39417 - 0.227254i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (0.979 + 4.90i)T \) |
| 7 | \( 1 + (-3.48 + 6.07i)T \) |
| good | 2 | \( 1 - 0.702iT - 4T^{2} \) |
| 11 | \( 1 - 5.39T + 121T^{2} \) |
| 13 | \( 1 - 12.5T + 169T^{2} \) |
| 17 | \( 1 + 8.14T + 289T^{2} \) |
| 19 | \( 1 + 1.94iT - 361T^{2} \) |
| 23 | \( 1 + 11.4iT - 529T^{2} \) |
| 29 | \( 1 + 44.2T + 841T^{2} \) |
| 31 | \( 1 - 50.4iT - 961T^{2} \) |
| 37 | \( 1 + 41.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.13iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 24.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 88.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 46.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 47.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 104. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 52.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 74.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 140.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 33.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 120.T + 9.40e3T^{2} \) |
| 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
−13.40763476093349526774745032492, −12.30965020616038527675021920363, −11.30755096878231890098992368039, −10.63553857958541174086196498441, −8.961628631248239803350114140906, −7.77855214277411105893074720779, −6.69268341678546758024374171596, −5.44940443950209280609428077314, −4.06529695700873666035573517650, −1.36909942464622270487531142915,
2.02623012757407563440984459122, 3.68280425842529100745979597652, 5.77821519823284938799005432045, 6.64969859414088555454659584164, 7.88179249561490169285637484161, 9.519143896961562290073647908488, 10.88217538473037062169700194765, 11.34938941540555813944748083550, 12.10251636262916561591613949787, 13.45535292635812439877405592522
