| L(s) = 1 | − 8·3-s + 14·5-s − 7·7-s + 37·9-s + 28·11-s − 18·13-s − 112·15-s + 74·17-s − 80·19-s + 56·21-s − 112·23-s + 71·25-s − 80·27-s − 190·29-s + 72·31-s − 224·33-s − 98·35-s + 346·37-s + 144·39-s + 162·41-s + 412·43-s + 518·45-s + 24·47-s + 49·49-s − 592·51-s − 318·53-s + 392·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 1.25·5-s − 0.377·7-s + 1.37·9-s + 0.767·11-s − 0.384·13-s − 1.92·15-s + 1.05·17-s − 0.965·19-s + 0.581·21-s − 1.01·23-s + 0.567·25-s − 0.570·27-s − 1.21·29-s + 0.417·31-s − 1.18·33-s − 0.473·35-s + 1.53·37-s + 0.591·39-s + 0.617·41-s + 1.46·43-s + 1.71·45-s + 0.0744·47-s + 1/7·49-s − 1.62·51-s − 0.824·53-s + 0.961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.300096063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300096063\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 346 T + p^{3} T^{2} \) |
| 41 | \( 1 - 162 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 198 T + p^{3} T^{2} \) |
| 67 | \( 1 - 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 538 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
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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
−10.68110404470469713138497396098, −9.865121602379741208016881640457, −9.318434730005439330273818253184, −7.72755946432516071307993723838, −6.43430105389589548630759035622, −6.06406703968984331099217459328, −5.24820924992053017111546256553, −4.02272005881669529475470171957, −2.15638860139020940383649297873, −0.793020032249763518782586109287,
0.793020032249763518782586109287, 2.15638860139020940383649297873, 4.02272005881669529475470171957, 5.24820924992053017111546256553, 6.06406703968984331099217459328, 6.43430105389589548630759035622, 7.72755946432516071307993723838, 9.318434730005439330273818253184, 9.865121602379741208016881640457, 10.68110404470469713138497396098
