L(s) = 1 | − 2·5-s + 4·11-s + 2·13-s − 6·17-s + 4·19-s − 25-s + 2·29-s + 6·37-s + 2·41-s + 4·43-s − 6·53-s − 8·55-s − 12·59-s + 2·61-s − 4·65-s − 4·67-s + 6·73-s + 16·79-s + 12·83-s + 12·85-s − 14·89-s − 8·95-s − 18·97-s + 14·101-s + 8·103-s + 4·107-s − 18·109-s + ⋯ |
L(s) = 1 | − 0.894·5-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 + 0.986·37-s + 0.312·41-s + 0.609·43-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 + 1.80·79-s + 1.31·83-s + 1.30·85-s − 1.48·89-s − 0.820·95-s − 1.82·97-s + 1.39·101-s + 0.788·103-s + 0.386·107-s − 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.668048632\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.668048632\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
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
−7.906821987114438610486627525496, −7.30073527958403832611391139576, −6.49804351404500129234723165210, −6.06370932399531155890837002587, −4.93184851224140584283722463407, −4.22946109432700828249314767474, −3.72933010810426492886781586543, −2.83876601704409626449741798211, −1.70938992598143604879955116667, −0.67392178676454816034771659294,
0.67392178676454816034771659294, 1.70938992598143604879955116667, 2.83876601704409626449741798211, 3.72933010810426492886781586543, 4.22946109432700828249314767474, 4.93184851224140584283722463407, 6.06370932399531155890837002587, 6.49804351404500129234723165210, 7.30073527958403832611391139576, 7.906821987114438610486627525496