| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s − 8-s + 9-s + 4·10-s − 4·11-s + 12-s − 4·13-s − 4·15-s + 16-s − 18-s − 4·19-s − 4·20-s + 4·22-s − 24-s + 11·25-s + 4·26-s + 27-s + 2·29-s + 4·30-s − 8·31-s − 32-s − 4·33-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s + 0.288·12-s − 1.10·13-s − 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.894·20-s + 0.852·22-s − 0.204·24-s + 11/5·25-s + 0.784·26-s + 0.192·27-s + 0.371·29-s + 0.730·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−11.10546510557791154808322951719, −10.47310858692444357093438177285, −9.234982336494788048482584951326, −8.236567804687403362610493554923, −7.69885140727101440087412366746, −6.94059953288016050503047892713, −5.02209723965846660973920127041, −3.77980943086478052006597909549, −2.52889971936759396667083662829, 0,
2.52889971936759396667083662829, 3.77980943086478052006597909549, 5.02209723965846660973920127041, 6.94059953288016050503047892713, 7.69885140727101440087412366746, 8.236567804687403362610493554923, 9.234982336494788048482584951326, 10.47310858692444357093438177285, 11.10546510557791154808322951719
