L(s) = 1 | − 2-s + 3-s + 4-s − 2.56·5-s − 6-s − 8-s + 9-s + 2.56·10-s + 3·11-s + 12-s − 3.12·13-s − 2.56·15-s + 16-s − 4.68·17-s − 18-s − 3.12·19-s − 2.56·20-s − 3·22-s − 23-s − 24-s + 1.56·25-s + 3.12·26-s + 27-s + 0.123·29-s + 2.56·30-s − 1.68·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.14·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.810·10-s + 0.904·11-s + 0.288·12-s − 0.866·13-s − 0.661·15-s + 0.250·16-s − 1.13·17-s − 0.235·18-s − 0.716·19-s − 0.572·20-s − 0.639·22-s − 0.208·23-s − 0.204·24-s + 0.312·25-s + 0.612·26-s + 0.192·27-s + 0.0228·29-s + 0.467·30-s − 0.302·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9693410771\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9693410771\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2.56T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 29 | \( 1 - 0.123T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 - 0.876T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 0.438T + 47T^{2} \) |
| 53 | \( 1 + 0.561T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 0.438T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6.56T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.032554515518659778400374986567, −7.41884570420309823927071423915, −6.83297769937344626833298449296, −6.19306638974892789918698850794, −4.94615080311815404792922685309, −4.16430401167084396760045025685, −3.64938350874356921009784311943, −2.59241444504186785061691936598, −1.85149492747534790548926656193, −0.52958996208150843629863591025,
0.52958996208150843629863591025, 1.85149492747534790548926656193, 2.59241444504186785061691936598, 3.64938350874356921009784311943, 4.16430401167084396760045025685, 4.94615080311815404792922685309, 6.19306638974892789918698850794, 6.83297769937344626833298449296, 7.41884570420309823927071423915, 8.032554515518659778400374986567