| L(s) = 1 | + 5·5-s + 30·7-s − 50·11-s − 88·13-s − 74·17-s + 140·19-s − 80·23-s + 25·25-s + 234·29-s + 150·35-s + 116·37-s + 72·41-s + 280·43-s − 120·47-s + 557·49-s + 498·53-s − 250·55-s + 870·59-s + 650·61-s − 440·65-s + 420·67-s − 1.02e3·71-s − 322·73-s − 1.50e3·77-s + 160·79-s + 980·83-s − 370·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.61·7-s − 1.37·11-s − 1.87·13-s − 1.05·17-s + 1.69·19-s − 0.725·23-s + 1/5·25-s + 1.49·29-s + 0.724·35-s + 0.515·37-s + 0.274·41-s + 0.993·43-s − 0.372·47-s + 1.62·49-s + 1.29·53-s − 0.612·55-s + 1.91·59-s + 1.36·61-s − 0.839·65-s + 0.765·67-s − 1.70·71-s − 0.516·73-s − 2.22·77-s + 0.227·79-s + 1.29·83-s − 0.472·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.399892718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399892718\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 30 T + p^{3} T^{2} \) |
| 11 | \( 1 + 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 88 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 116 T + p^{3} T^{2} \) |
| 41 | \( 1 - 72 T + p^{3} T^{2} \) |
| 43 | \( 1 - 280 T + p^{3} T^{2} \) |
| 47 | \( 1 + 120 T + p^{3} T^{2} \) |
| 53 | \( 1 - 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 870 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 - 420 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 980 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1124 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1114 T + p^{3} T^{2} \) |
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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
−9.183568322421672675622707948291, −8.163125776027674129353610311879, −7.67667672076264051969944230031, −6.92925142444115562582730730012, −5.49556062424036617247306294141, −5.08404376707738621745111316147, −4.36677504441589569930927826453, −2.64066800763490306765733026788, −2.17097995235642993453165987627, −0.75518088363383086160832809223,
0.75518088363383086160832809223, 2.17097995235642993453165987627, 2.64066800763490306765733026788, 4.36677504441589569930927826453, 5.08404376707738621745111316147, 5.49556062424036617247306294141, 6.92925142444115562582730730012, 7.67667672076264051969944230031, 8.163125776027674129353610311879, 9.183568322421672675622707948291
