| L(s) = 1 | − 2·2-s + 4·4-s + 2·5-s − 8·8-s − 4·10-s + 8·11-s + 42·13-s + 16·16-s − 2·17-s + 124·19-s + 8·20-s − 16·22-s − 76·23-s − 121·25-s − 84·26-s − 254·29-s + 72·31-s − 32·32-s + 4·34-s + 398·37-s − 248·38-s − 16·40-s + 462·41-s + 212·43-s + 32·44-s + 152·46-s − 264·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.178·5-s − 0.353·8-s − 0.126·10-s + 0.219·11-s + 0.896·13-s + 1/4·16-s − 0.0285·17-s + 1.49·19-s + 0.0894·20-s − 0.155·22-s − 0.689·23-s − 0.967·25-s − 0.633·26-s − 1.62·29-s + 0.417·31-s − 0.176·32-s + 0.0201·34-s + 1.76·37-s − 1.05·38-s − 0.0632·40-s + 1.75·41-s + 0.751·43-s + 0.109·44-s + 0.487·46-s − 0.819·47-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
| \(L(2)\) |
\(\approx\) |
\(1.587788402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587788402\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 76 T + p^{3} T^{2} \) |
| 29 | \( 1 + 254 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 462 T + p^{3} T^{2} \) |
| 43 | \( 1 - 212 T + p^{3} T^{2} \) |
| 47 | \( 1 + 264 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 772 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 - 236 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1190 T + p^{3} T^{2} \) |
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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
−9.464968483560731200847417881034, −9.225740586997632855980256308969, −7.897735412848267713296049066202, −7.54340419523430447364457449954, −6.20993220105499188921119240447, −5.71372285280745959980201813751, −4.25216111617852756841148328623, −3.19117438164132085872473581154, −1.90607130691474740555659522609, −0.78773579670235468492975558552,
0.78773579670235468492975558552, 1.90607130691474740555659522609, 3.19117438164132085872473581154, 4.25216111617852756841148328623, 5.71372285280745959980201813751, 6.20993220105499188921119240447, 7.54340419523430447364457449954, 7.897735412848267713296049066202, 9.225740586997632855980256308969, 9.464968483560731200847417881034
