| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 11-s − 12-s − 2·15-s + 16-s + 6·17-s − 18-s + 2·20-s + 22-s + 4·23-s + 24-s − 25-s − 27-s + 6·29-s + 2·30-s − 4·31-s − 32-s + 33-s − 6·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.365·30-s − 0.718·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.80914641646317, −16.43369239219527, −15.78529741850852, −15.19192778695079, −14.45513675290881, −14.02086789859410, −13.17197585094506, −12.78572549535060, −12.01317989011360, −11.55891583548420, −10.84664643376246, −10.14787272199259, −9.961815927772440, −9.296611057349563, −8.517884980864016, −7.926474242724361, −7.212105058052368, −6.537614721440746, −6.004398862039332, −5.251458434767349, −4.841529390259386, −3.516001517083775, −2.907982770143993, −1.791801190991859, −1.243886444881757, 0,
1.243886444881757, 1.791801190991859, 2.907982770143993, 3.516001517083775, 4.841529390259386, 5.251458434767349, 6.004398862039332, 6.537614721440746, 7.212105058052368, 7.926474242724361, 8.517884980864016, 9.296611057349563, 9.961815927772440, 10.14787272199259, 10.84664643376246, 11.55891583548420, 12.01317989011360, 12.78572549535060, 13.17197585094506, 14.02086789859410, 14.45513675290881, 15.19192778695079, 15.78529741850852, 16.43369239219527, 16.80914641646317
