L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3.04·7-s − 8-s + 9-s + 10-s − 1.13·11-s − 12-s + 3.04·14-s + 15-s + 16-s + 2.69·17-s − 18-s − 1.55·19-s − 20-s + 3.04·21-s + 1.13·22-s + 5.40·23-s + 24-s + 25-s − 27-s − 3.04·28-s − 6.15·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.15·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.342·11-s − 0.288·12-s + 0.814·14-s + 0.258·15-s + 0.250·16-s + 0.652·17-s − 0.235·18-s − 0.356·19-s − 0.223·20-s + 0.665·21-s + 0.242·22-s + 1.12·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.576·28-s − 1.14·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 + 6.15T + 29T^{2} \) |
| 31 | \( 1 - 0.246T + 31T^{2} \) |
| 37 | \( 1 - 0.554T + 37T^{2} \) |
| 41 | \( 1 + 8.26T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 8.57T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 1.93T + 97T^{2} \) |
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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
−7.893531403007937220462835591154, −6.97832734829668655832231845459, −6.75446063912457778681237804636, −5.73143900944716364369844551479, −5.17818296966509723263788229667, −3.96452721617548886795754718796, −3.29024831008077842920425188978, −2.33159237780172804737933699493, −0.991237008844185505469845454745, 0,
0.991237008844185505469845454745, 2.33159237780172804737933699493, 3.29024831008077842920425188978, 3.96452721617548886795754718796, 5.17818296966509723263788229667, 5.73143900944716364369844551479, 6.75446063912457778681237804636, 6.97832734829668655832231845459, 7.893531403007937220462835591154