L(s) = 1 | − 1.36·2-s + 3-s − 0.124·4-s − 5-s − 1.36·6-s + 7-s + 2.90·8-s + 9-s + 1.36·10-s + 11-s − 0.124·12-s − 2.27·13-s − 1.36·14-s − 15-s − 3.73·16-s + 2.48·17-s − 1.36·18-s + 3.12·19-s + 0.124·20-s + 21-s − 1.36·22-s − 1.61·23-s + 2.90·24-s + 25-s + 3.12·26-s + 27-s − 0.124·28-s + ⋯ |
L(s) = 1 | − 0.968·2-s + 0.577·3-s − 0.0621·4-s − 0.447·5-s − 0.559·6-s + 0.377·7-s + 1.02·8-s + 0.333·9-s + 0.433·10-s + 0.301·11-s − 0.0358·12-s − 0.632·13-s − 0.366·14-s − 0.258·15-s − 0.933·16-s + 0.603·17-s − 0.322·18-s + 0.716·19-s + 0.0277·20-s + 0.218·21-s − 0.291·22-s − 0.336·23-s + 0.593·24-s + 0.200·25-s + 0.612·26-s + 0.192·27-s − 0.0234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.051721482\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.051721482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 + 9.01T + 29T^{2} \) |
| 31 | \( 1 - 9.01T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 4.35T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.11T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.633473511617270897311394243116, −9.028564113322997530590823594862, −8.103778097802350158094198806597, −7.69842941673830969163393393011, −6.91285535309201583216136636759, −5.46399584260304120802708049630, −4.48525786874413755529092555832, −3.59360221830395648810042673571, −2.21103285528245149504466251361, −0.904439964223657288291777772045,
0.904439964223657288291777772045, 2.21103285528245149504466251361, 3.59360221830395648810042673571, 4.48525786874413755529092555832, 5.46399584260304120802708049630, 6.91285535309201583216136636759, 7.69842941673830969163393393011, 8.103778097802350158094198806597, 9.028564113322997530590823594862, 9.633473511617270897311394243116