L(s) = 1 | + 2-s − 1.80·3-s + 4-s − 0.445·5-s − 1.80·6-s + 8-s + 2.24·9-s − 0.445·10-s − 1.80·12-s − 1.80·13-s + 0.801·15-s + 16-s + 2.24·18-s + 2·19-s − 0.445·20-s + 1.24·23-s − 1.80·24-s − 0.801·25-s − 1.80·26-s − 2.24·27-s + 0.801·30-s − 0.445·31-s + 32-s + 2.24·36-s + 1.24·37-s + 2·38-s + 3.24·39-s + ⋯ |
L(s) = 1 | + 2-s − 1.80·3-s + 4-s − 0.445·5-s − 1.80·6-s + 8-s + 2.24·9-s − 0.445·10-s − 1.80·12-s − 1.80·13-s + 0.801·15-s + 16-s + 2.24·18-s + 2·19-s − 0.445·20-s + 1.24·23-s − 1.80·24-s − 0.801·25-s − 1.80·26-s − 2.24·27-s + 0.801·30-s − 0.445·31-s + 32-s + 2.24·36-s + 1.24·37-s + 2·38-s + 3.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.207955621\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.207955621\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 641 | \( 1 - T \) |
good | 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 + 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 2T + T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - 1.24T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.445T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.480471276610061054638770748189, −7.66135527390400657462288917242, −7.34586361052291921187671365601, −6.67854557656562783428738425182, −5.66641119706934048466594407279, −5.21089700598233530147779835995, −4.65087159264903220869057912905, −3.70577974528177021689374953445, −2.49289740971566458604209512827, −0.997091823585141909913860657109,
0.997091823585141909913860657109, 2.49289740971566458604209512827, 3.70577974528177021689374953445, 4.65087159264903220869057912905, 5.21089700598233530147779835995, 5.66641119706934048466594407279, 6.67854557656562783428738425182, 7.34586361052291921187671365601, 7.66135527390400657462288917242, 9.480471276610061054638770748189