| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s + 25-s − 2·26-s + 6·29-s − 8·31-s + 32-s + 6·34-s + 2·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s + 50-s − 2·52-s + 6·53-s + 6·58-s + 10·61-s − 8·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.141·50-s − 0.277·52-s + 0.824·53-s + 0.787·58-s + 1.28·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.925820309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.925820309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.145462184294923465930650647398, −7.51828953604385192982905489512, −6.97913667513999229030385049043, −6.03795573121967783311486434379, −5.26570153655431196939565550355, −4.75701556135232590446975582633, −3.63359445531324431774910445192, −3.21826656477469471981988966496, −2.10133345305515366205304847130, −0.879817280069523657260066044025,
0.879817280069523657260066044025, 2.10133345305515366205304847130, 3.21826656477469471981988966496, 3.63359445531324431774910445192, 4.75701556135232590446975582633, 5.26570153655431196939565550355, 6.03795573121967783311486434379, 6.97913667513999229030385049043, 7.51828953604385192982905489512, 8.145462184294923465930650647398
