L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s − 2·9-s + 3·11-s + 2·12-s + 5·13-s + 15-s + 4·16-s − 3·17-s − 2·19-s + 2·20-s − 21-s − 6·23-s + 25-s + 5·27-s − 2·28-s + 4·31-s − 3·33-s − 35-s + 4·36-s − 2·37-s − 5·39-s + 12·41-s + 10·43-s − 6·44-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s + 0.447·20-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s + 0.718·31-s − 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174746062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174746062\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.94954252094320, −14.66991459123645, −13.98264172251917, −13.69926595765268, −13.09498396998376, −12.37140033018981, −11.96633820186615, −11.46272576477860, −10.80815513511686, −10.59039216224496, −9.652045618344440, −9.014142244003701, −8.702360398195978, −8.160597599707735, −7.633028705709818, −6.667022850720948, −6.026788921832984, −5.850213979514909, −4.899622908166825, −4.268621302938483, −3.970782429389669, −3.210874340845151, −2.192030850024015, −1.160564968745801, −0.5135114011224828,
0.5135114011224828, 1.160564968745801, 2.192030850024015, 3.210874340845151, 3.970782429389669, 4.268621302938483, 4.899622908166825, 5.850213979514909, 6.026788921832984, 6.667022850720948, 7.633028705709818, 8.160597599707735, 8.702360398195978, 9.014142244003701, 9.652045618344440, 10.59039216224496, 10.80815513511686, 11.46272576477860, 11.96633820186615, 12.37140033018981, 13.09498396998376, 13.69926595765268, 13.98264172251917, 14.66991459123645, 14.94954252094320