L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 2·13-s + 14-s + 16-s + 6·17-s − 19-s + 6·23-s − 2·26-s + 28-s + 6·29-s + 5·31-s + 32-s + 6·34-s + 7·37-s − 38-s − 12·41-s − 11·43-s + 6·46-s + 12·47-s − 6·49-s − 2·52-s + 56-s + 6·58-s + 6·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.229·19-s + 1.25·23-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s + 1.15·37-s − 0.162·38-s − 1.87·41-s − 1.67·43-s + 0.884·46-s + 1.75·47-s − 6/7·49-s − 0.277·52-s + 0.133·56-s + 0.787·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.835323254\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.835323254\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−9.846703955492822247090368779764, −8.656989470766764405538809884479, −7.908767328265442465054092066225, −7.06887111301165634585693605346, −6.25534581212311513209126682009, −5.20821295620093410550345915508, −4.69361595782645751634101798872, −3.48129308371563671231369503813, −2.62316968096441052690749315475, −1.22191673569958869404887128849,
1.22191673569958869404887128849, 2.62316968096441052690749315475, 3.48129308371563671231369503813, 4.69361595782645751634101798872, 5.20821295620093410550345915508, 6.25534581212311513209126682009, 7.06887111301165634585693605346, 7.908767328265442465054092066225, 8.656989470766764405538809884479, 9.846703955492822247090368779764