L(s) = 1 | − 2·5-s − 7-s + 4·11-s + 2·13-s + 6·17-s − 4·19-s − 25-s − 2·29-s + 2·35-s − 6·37-s − 2·41-s + 4·43-s + 49-s + 6·53-s − 8·55-s + 12·59-s + 2·61-s − 4·65-s − 4·67-s − 6·73-s − 4·77-s − 16·79-s − 12·83-s − 12·85-s + 14·89-s − 2·91-s + 8·95-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.371·29-s + 0.338·35-s − 0.986·37-s − 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.07·55-s + 1.56·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s − 0.702·73-s − 0.455·77-s − 1.80·79-s − 1.31·83-s − 1.30·85-s + 1.48·89-s − 0.209·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.560316622\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.560316622\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.597097830639437040754263866908, −7.62195349072755325091614948497, −7.09250692106396557822287287333, −6.22411641525239374249127245960, −5.60865912599068037214391350294, −4.45044632349803314255794033962, −3.76538094002949404605955440773, −3.26988486528469799014112835839, −1.87090915832039374223761593463, −0.72892880515889973617029791676,
0.72892880515889973617029791676, 1.87090915832039374223761593463, 3.26988486528469799014112835839, 3.76538094002949404605955440773, 4.45044632349803314255794033962, 5.60865912599068037214391350294, 6.22411641525239374249127245960, 7.09250692106396557822287287333, 7.62195349072755325091614948497, 8.597097830639437040754263866908