| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s − 2·9-s − 10-s − 4·11-s − 12-s + 4·13-s + 2·14-s − 15-s + 16-s − 3·17-s + 2·18-s − 19-s + 20-s + 2·21-s + 4·22-s + 24-s + 25-s − 4·26-s + 5·27-s − 2·28-s + 8·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.436·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.962·27-s − 0.377·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−8.092780463171113875255565807394, −6.92772845799665143843220258778, −6.45242391684661947365057416059, −5.81227294688163494427466506114, −5.19607813943749389457142853282, −4.10915396600608543603540796418, −2.92620584550897948112824315949, −2.45883209203783844824642984584, −1.05331142214207878850925673042, 0,
1.05331142214207878850925673042, 2.45883209203783844824642984584, 2.92620584550897948112824315949, 4.10915396600608543603540796418, 5.19607813943749389457142853282, 5.81227294688163494427466506114, 6.45242391684661947365057416059, 6.92772845799665143843220258778, 8.092780463171113875255565807394
