| L(s) = 1 | + 2·3-s + 9·5-s − 31·7-s − 23·9-s − 57·11-s + 52·13-s + 18·15-s + 69·17-s − 19·19-s − 62·21-s − 72·23-s − 44·25-s − 100·27-s + 150·29-s + 32·31-s − 114·33-s − 279·35-s + 226·37-s + 104·39-s − 258·41-s + 67·43-s − 207·45-s + 579·47-s + 618·49-s + 138·51-s + 432·53-s − 513·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.804·5-s − 1.67·7-s − 0.851·9-s − 1.56·11-s + 1.10·13-s + 0.309·15-s + 0.984·17-s − 0.229·19-s − 0.644·21-s − 0.652·23-s − 0.351·25-s − 0.712·27-s + 0.960·29-s + 0.185·31-s − 0.601·33-s − 1.34·35-s + 1.00·37-s + 0.427·39-s − 0.982·41-s + 0.237·43-s − 0.685·45-s + 1.79·47-s + 1.80·49-s + 0.378·51-s + 1.11·53-s − 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.681310875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681310875\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 31 T + p^{3} T^{2} \) |
| 11 | \( 1 + 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 69 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 32 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 258 T + p^{3} T^{2} \) |
| 43 | \( 1 - 67 T + p^{3} T^{2} \) |
| 47 | \( 1 - 579 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 330 T + p^{3} T^{2} \) |
| 61 | \( 1 - 13 T + p^{3} T^{2} \) |
| 67 | \( 1 - 856 T + p^{3} T^{2} \) |
| 71 | \( 1 - 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1424 T + p^{3} T^{2} \) |
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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
−9.453816499055840731714963848316, −8.558096012505536675416376489700, −7.892633949483812564997519110026, −6.72476968468592640044011233852, −5.83847548147430368548314356752, −5.55057924567218571015087346590, −3.87066647141817094262092517610, −2.98043041281799238156012586778, −2.35040027963981661612943068714, −0.61749852112491609360435525372,
0.61749852112491609360435525372, 2.35040027963981661612943068714, 2.98043041281799238156012586778, 3.87066647141817094262092517610, 5.55057924567218571015087346590, 5.83847548147430368548314356752, 6.72476968468592640044011233852, 7.892633949483812564997519110026, 8.558096012505536675416376489700, 9.453816499055840731714963848316
