| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 2·13-s − 14-s + 16-s − 6·17-s + 4·19-s − 20-s + 22-s − 4·23-s + 25-s + 2·26-s − 28-s + 2·29-s + 8·31-s + 32-s − 6·34-s + 35-s − 10·37-s + 4·38-s − 40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.169·35-s − 1.64·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781522467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781522467\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.019267275988769803363880152362, −6.91736496796436123419820657606, −6.66024671863858501032671715532, −5.85030921980011737885080256856, −5.04878326356972338900087223647, −4.25869365448845163691866991707, −3.71804717091682032396061659043, −2.87866146590013093205329187248, −2.00354921884331190027247582014, −0.75919813140686390122392002353,
0.75919813140686390122392002353, 2.00354921884331190027247582014, 2.87866146590013093205329187248, 3.71804717091682032396061659043, 4.25869365448845163691866991707, 5.04878326356972338900087223647, 5.85030921980011737885080256856, 6.66024671863858501032671715532, 6.91736496796436123419820657606, 8.019267275988769803363880152362
