| L(s) = 1 | + 2-s − 2.52·3-s + 4-s − 2.45·5-s − 2.52·6-s − 2.79·7-s + 8-s + 3.35·9-s − 2.45·10-s − 1.67·11-s − 2.52·12-s + 6.34·13-s − 2.79·14-s + 6.19·15-s + 16-s + 4.96·17-s + 3.35·18-s − 2.45·20-s + 7.04·21-s − 1.67·22-s − 2.49·23-s − 2.52·24-s + 1.04·25-s + 6.34·26-s − 0.884·27-s − 2.79·28-s + 5.93·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.45·3-s + 0.5·4-s − 1.09·5-s − 1.02·6-s − 1.05·7-s + 0.353·8-s + 1.11·9-s − 0.777·10-s − 0.506·11-s − 0.727·12-s + 1.75·13-s − 0.746·14-s + 1.60·15-s + 0.250·16-s + 1.20·17-s + 0.789·18-s − 0.549·20-s + 1.53·21-s − 0.357·22-s − 0.521·23-s − 0.514·24-s + 0.209·25-s + 1.24·26-s − 0.170·27-s − 0.527·28-s + 1.10·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.016536338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.016536338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 - 0.550T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 0.745T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 9.33T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 6.90T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−10.59025843226846029669737145984, −10.03756417314902156290134230636, −8.477611957831713683391801872087, −7.58989442930225679912534136349, −6.41486782981149941317317440355, −6.09055955647936607315887004263, −5.03640209850765101757511204887, −3.97475101148410634055362912717, −3.17358334594221854836744868679, −0.821622696110350595198846364947,
0.821622696110350595198846364947, 3.17358334594221854836744868679, 3.97475101148410634055362912717, 5.03640209850765101757511204887, 6.09055955647936607315887004263, 6.41486782981149941317317440355, 7.58989442930225679912534136349, 8.477611957831713683391801872087, 10.03756417314902156290134230636, 10.59025843226846029669737145984
