| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.64i·7-s − 2.82·8-s + 6.61i·11-s − 6.21i·13-s + 3.74i·14-s + 4.00·16-s − 6.35·17-s + 23.5·19-s − 9.35i·22-s − 12.8·23-s + 8.79i·26-s − 5.29i·28-s + 21.7i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.377i·7-s − 0.353·8-s + 0.601i·11-s − 0.478i·13-s + 0.267i·14-s + 0.250·16-s − 0.374·17-s + 1.24·19-s − 0.425i·22-s − 0.559·23-s + 0.338i·26-s − 0.188i·28-s + 0.748i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4066258429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4066258429\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 - 6.61iT - 121T^{2} \) |
| 13 | \( 1 + 6.21iT - 169T^{2} \) |
| 17 | \( 1 + 6.35T + 289T^{2} \) |
| 19 | \( 1 - 23.5T + 361T^{2} \) |
| 23 | \( 1 + 12.8T + 529T^{2} \) |
| 29 | \( 1 - 21.7iT - 841T^{2} \) |
| 31 | \( 1 + 39.5T + 961T^{2} \) |
| 37 | \( 1 + 56.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 20.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 40.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 25.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 0.743T + 3.72e3T^{2} \) |
| 67 | \( 1 + 68.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 56.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 22.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 6.08iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 135. iT - 9.40e3T^{2} \) |
| 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
−8.174179402931194118544931743724, −7.32077310879493151664465648081, −7.12392868010942288838115968454, −5.92282928249047715535270097262, −5.28313573281689750884045822842, −4.19028737072383947281483168108, −3.32156163668369228502670988144, −2.27293842551504971355561496794, −1.28447313279649466651582452895, −0.12486267813143189706710405152,
1.13045723820270065126075568732, 2.16871132396036538433979106335, 3.10954685473141110654460277968, 4.02808822804492667425515765779, 5.17632705864766854873469523782, 5.88271336545679732998567281401, 6.67625727891361176600395307261, 7.42769450177673006388951686147, 8.186916433967964786377370123197, 8.819152354515147317129525026850
