L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 14-s + 15-s + 16-s − 6·17-s − 18-s + 4·19-s + 20-s − 21-s − 24-s + 25-s + 27-s − 28-s − 6·29-s − 30-s + 4·31-s − 32-s + 6·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.23136638120599, −14.80926655015797, −14.05567424414242, −13.63172274204289, −13.19031209886079, −12.64478574609744, −11.95253499089684, −11.44286700632686, −10.84147082622908, −10.26768010207005, −9.794951021802713, −9.294329714389412, −8.709238748015368, −8.503376143289204, −7.531252309232622, −7.114278348479379, −6.720009643391791, −5.776436956456613, −5.516317705299522, −4.400182103324877, −3.961288316065706, −2.987563783948734, −2.557521691542226, −1.839289640477686, −1.053626251642212, 0,
1.053626251642212, 1.839289640477686, 2.557521691542226, 2.987563783948734, 3.961288316065706, 4.400182103324877, 5.516317705299522, 5.776436956456613, 6.720009643391791, 7.114278348479379, 7.531252309232622, 8.503376143289204, 8.709238748015368, 9.294329714389412, 9.794951021802713, 10.26768010207005, 10.84147082622908, 11.44286700632686, 11.95253499089684, 12.64478574609744, 13.19031209886079, 13.63172274204289, 14.05567424414242, 14.80926655015797, 15.23136638120599