| L(s) = 1 | − 2·2-s + 4·4-s + 34·7-s − 8·8-s − 48·11-s + 70·13-s − 68·14-s + 16·16-s + 27·17-s + 119·19-s + 96·22-s + 51·23-s − 140·26-s + 136·28-s − 30·29-s − 133·31-s − 32·32-s − 54·34-s − 218·37-s − 238·38-s + 156·41-s + 88·43-s − 192·44-s − 102·46-s + 516·47-s + 813·49-s + 280·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.83·7-s − 0.353·8-s − 1.31·11-s + 1.49·13-s − 1.29·14-s + 1/4·16-s + 0.385·17-s + 1.43·19-s + 0.930·22-s + 0.462·23-s − 1.05·26-s + 0.917·28-s − 0.192·29-s − 0.770·31-s − 0.176·32-s − 0.272·34-s − 0.968·37-s − 1.01·38-s + 0.594·41-s + 0.312·43-s − 0.657·44-s − 0.326·46-s + 1.60·47-s + 2.37·49-s + 0.746·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.154126409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154126409\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 19 | \( 1 - 119 T + p^{3} T^{2} \) |
| 23 | \( 1 - 51 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 133 T + p^{3} T^{2} \) |
| 37 | \( 1 + 218 T + p^{3} T^{2} \) |
| 41 | \( 1 - 156 T + p^{3} T^{2} \) |
| 43 | \( 1 - 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 516 T + p^{3} T^{2} \) |
| 53 | \( 1 - 639 T + p^{3} T^{2} \) |
| 59 | \( 1 + 654 T + p^{3} T^{2} \) |
| 61 | \( 1 - 461 T + p^{3} T^{2} \) |
| 67 | \( 1 + 182 T + p^{3} T^{2} \) |
| 71 | \( 1 - 900 T + p^{3} T^{2} \) |
| 73 | \( 1 + 704 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1375 T + p^{3} T^{2} \) |
| 83 | \( 1 + 915 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1116 T + p^{3} T^{2} \) |
| 97 | \( 1 - 16 T + p^{3} T^{2} \) |
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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
−8.967369679141686204191911447696, −8.468165483209207599618407466535, −7.66602866119303579933611511508, −7.24490973541284008571569762385, −5.64499348959912562723210833175, −5.34584134594257413075655205757, −4.08273837494305759264712770041, −2.85725710273124216864700745587, −1.68876263435477727752157884566, −0.884715216045029491946563229060,
0.884715216045029491946563229060, 1.68876263435477727752157884566, 2.85725710273124216864700745587, 4.08273837494305759264712770041, 5.34584134594257413075655205757, 5.64499348959912562723210833175, 7.24490973541284008571569762385, 7.66602866119303579933611511508, 8.468165483209207599618407466535, 8.967369679141686204191911447696
