L(s) = 1 | + 2-s + 3-s + 4-s + 0.183·5-s + 6-s + 8-s + 9-s + 0.183·10-s + 3.08·11-s + 12-s − 5.85·13-s + 0.183·15-s + 16-s − 6.36·17-s + 18-s − 3.41·19-s + 0.183·20-s + 3.08·22-s − 23-s + 24-s − 4.96·25-s − 5.85·26-s + 27-s − 4.68·29-s + 0.183·30-s − 3.41·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0822·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.0581·10-s + 0.931·11-s + 0.288·12-s − 1.62·13-s + 0.0475·15-s + 0.250·16-s − 1.54·17-s + 0.235·18-s − 0.783·19-s + 0.0411·20-s + 0.658·22-s − 0.208·23-s + 0.204·24-s − 0.993·25-s − 1.14·26-s + 0.192·27-s − 0.870·29-s + 0.0335·30-s − 0.613·31-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 0.183T + 5T^{2} \) |
| 11 | \( 1 - 3.08T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 + 1.79T + 53T^{2} \) |
| 59 | \( 1 + 7.91T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 + 1.53T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 + 9.08T + 79T^{2} \) |
| 83 | \( 1 + 0.0175T + 83T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
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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
−7.48357626926373796786526087565, −6.89770901503022131151010714270, −6.29369381339892336656965989326, −5.42755670756836518958077068796, −4.50924206310433909335514475136, −4.14694235637468820872193577753, −3.22910151587300674167941918842, −2.23617440505533557655414793442, −1.82867776158053580676924648167, 0,
1.82867776158053580676924648167, 2.23617440505533557655414793442, 3.22910151587300674167941918842, 4.14694235637468820872193577753, 4.50924206310433909335514475136, 5.42755670756836518958077068796, 6.29369381339892336656965989326, 6.89770901503022131151010714270, 7.48357626926373796786526087565