L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 4·11-s + 12-s − 4·13-s + 16-s + 18-s + 4·19-s − 4·22-s + 24-s − 4·26-s + 27-s + 2·29-s + 8·31-s + 32-s − 4·33-s + 36-s + 6·37-s + 4·38-s − 4·39-s − 4·43-s − 4·44-s + 8·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.852·22-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.986·37-s + 0.648·38-s − 0.640·39-s − 0.609·43-s − 0.603·44-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.777683404\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.777683404\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 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
−7.79653315257239980649946962627, −7.29197164616250683867458907702, −6.54353825482436203942086711241, −5.62609399472295508440674319571, −5.00728521418144527397618618017, −4.43147582495336180982420814015, −3.46416758656936674716250658796, −2.67089709547108972531049107608, −2.26571717286938863297527061889, −0.841600108109119125083090617074,
0.841600108109119125083090617074, 2.26571717286938863297527061889, 2.67089709547108972531049107608, 3.46416758656936674716250658796, 4.43147582495336180982420814015, 5.00728521418144527397618618017, 5.62609399472295508440674319571, 6.54353825482436203942086711241, 7.29197164616250683867458907702, 7.79653315257239980649946962627