L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s − 3·11-s − 4·13-s + 14-s + 16-s − 3·17-s − 3·18-s − 3·22-s − 8·23-s − 4·26-s + 28-s − 3·29-s − 7·31-s + 32-s − 3·34-s − 3·36-s − 37-s + 11·41-s + 11·43-s − 3·44-s − 8·46-s + 4·47-s − 6·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s − 0.904·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.639·22-s − 1.66·23-s − 0.784·26-s + 0.188·28-s − 0.557·29-s − 1.25·31-s + 0.176·32-s − 0.514·34-s − 1/2·36-s − 0.164·37-s + 1.71·41-s + 1.67·43-s − 0.452·44-s − 1.17·46-s + 0.583·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.854057692422553285263194591852, −7.73446756997252531037083714905, −7.50239233224545552250828004199, −6.15186997911867805479601305524, −5.61522714744380613862444046217, −4.78501568576043254870076108978, −3.93554524857205013821612825315, −2.71184803712278664440731522878, −2.10157444826247915682076467458, 0,
2.10157444826247915682076467458, 2.71184803712278664440731522878, 3.93554524857205013821612825315, 4.78501568576043254870076108978, 5.61522714744380613862444046217, 6.15186997911867805479601305524, 7.50239233224545552250828004199, 7.73446756997252531037083714905, 8.854057692422553285263194591852