| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 4·11-s + 2·13-s + 16-s − 8·17-s − 6·19-s − 20-s − 4·22-s + 4·23-s + 25-s − 2·26-s + 6·29-s + 4·31-s − 32-s + 8·34-s − 10·37-s + 6·38-s + 40-s − 4·41-s + 4·43-s + 4·44-s − 4·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.94·17-s − 1.37·19-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.37·34-s − 1.64·37-s + 0.973·38-s + 0.158·40-s − 0.624·41-s + 0.609·43-s + 0.603·44-s − 0.589·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.291052151464846112138820174379, −7.17868401996569353961157017022, −6.55888869746680490776532532380, −6.26950024478396089929254432559, −4.83213516401447160679471202136, −4.22772376308517161182928759426, −3.32483196694575894697168559689, −2.25505360853015328005927850296, −1.29930809210743868304411696124, 0,
1.29930809210743868304411696124, 2.25505360853015328005927850296, 3.32483196694575894697168559689, 4.22772376308517161182928759426, 4.83213516401447160679471202136, 6.26950024478396089929254432559, 6.55888869746680490776532532380, 7.17868401996569353961157017022, 8.291052151464846112138820174379
