| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 11-s + 12-s − 2·13-s − 16-s − 6·17-s + 18-s + 4·19-s + 22-s − 8·23-s + 3·24-s − 2·26-s − 27-s + 6·29-s − 8·31-s + 5·32-s − 33-s − 6·34-s − 36-s − 6·37-s + 4·38-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.213·22-s − 1.66·23-s + 0.612·24-s − 0.392·26-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.174·33-s − 1.02·34-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.83919049778719, −14.46580106001723, −13.94317739756052, −13.51833508952361, −12.93619986039425, −12.43923676475855, −12.05608434176916, −11.43659220209902, −11.11676159614207, −10.12795557420815, −9.914763386790766, −9.262160755927954, −8.628357840346500, −8.233442295709189, −7.248064622503887, −6.918565044374317, −6.166800035702735, −5.662206422569633, −5.162443278210798, −4.539638496495351, −4.017127495391137, −3.530970681151896, −2.556899111754063, −1.949970339725901, −0.7920259325669287, 0,
0.7920259325669287, 1.949970339725901, 2.556899111754063, 3.530970681151896, 4.017127495391137, 4.539638496495351, 5.162443278210798, 5.662206422569633, 6.166800035702735, 6.918565044374317, 7.248064622503887, 8.233442295709189, 8.628357840346500, 9.262160755927954, 9.914763386790766, 10.12795557420815, 11.11676159614207, 11.43659220209902, 12.05608434176916, 12.43923676475855, 12.93619986039425, 13.51833508952361, 13.94317739756052, 14.46580106001723, 14.83919049778719
