L(s) = 1 | + 2·2-s + 4·4-s + 6·5-s + 8·8-s + 12·10-s + 30·11-s + 53·13-s + 16·16-s + 84·17-s − 97·19-s + 24·20-s + 60·22-s − 84·23-s − 89·25-s + 106·26-s + 180·29-s + 179·31-s + 32·32-s + 168·34-s − 145·37-s − 194·38-s + 48·40-s − 126·41-s − 325·43-s + 120·44-s − 168·46-s + 366·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.536·5-s + 0.353·8-s + 0.379·10-s + 0.822·11-s + 1.13·13-s + 1/4·16-s + 1.19·17-s − 1.17·19-s + 0.268·20-s + 0.581·22-s − 0.761·23-s − 0.711·25-s + 0.799·26-s + 1.15·29-s + 1.03·31-s + 0.176·32-s + 0.847·34-s − 0.644·37-s − 0.828·38-s + 0.189·40-s − 0.479·41-s − 1.15·43-s + 0.411·44-s − 0.538·46-s + 1.13·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.219559458\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.219559458\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 53 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 97 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 180 T + p^{3} T^{2} \) |
| 31 | \( 1 - 179 T + p^{3} T^{2} \) |
| 37 | \( 1 + 145 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 325 T + p^{3} T^{2} \) |
| 47 | \( 1 - 366 T + p^{3} T^{2} \) |
| 53 | \( 1 - 768 T + p^{3} T^{2} \) |
| 59 | \( 1 - 264 T + p^{3} T^{2} \) |
| 61 | \( 1 - 818 T + p^{3} T^{2} \) |
| 67 | \( 1 + 523 T + p^{3} T^{2} \) |
| 71 | \( 1 - 342 T + p^{3} T^{2} \) |
| 73 | \( 1 + 43 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1171 T + p^{3} T^{2} \) |
| 83 | \( 1 - 810 T + p^{3} T^{2} \) |
| 89 | \( 1 - 600 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−10.05506942637359343655414795719, −8.787148522613480694708585065152, −8.142565341057299212836313481607, −6.85660579523683537053121006690, −6.18924324874339653487200823264, −5.48521086229103939987071304482, −4.24976690707695419636168562726, −3.51430818787618737544820718574, −2.20303142741957992740104388077, −1.09792233414450520517346022926,
1.09792233414450520517346022926, 2.20303142741957992740104388077, 3.51430818787618737544820718574, 4.24976690707695419636168562726, 5.48521086229103939987071304482, 6.18924324874339653487200823264, 6.85660579523683537053121006690, 8.142565341057299212836313481607, 8.787148522613480694708585065152, 10.05506942637359343655414795719