L(s) = 1 | + 1.41·2-s + 1.00·4-s + 1.41i·5-s + 2.00i·10-s + (0.707 + 0.707i)11-s − i·13-s − 0.999·16-s + 1.41i·20-s + (1.00 + 1.00i)22-s − 1.00·25-s − 1.41i·26-s − 1.41·32-s + 1.41·41-s + (0.707 + 0.707i)44-s − 1.41i·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.00·4-s + 1.41i·5-s + 2.00i·10-s + (0.707 + 0.707i)11-s − i·13-s − 0.999·16-s + 1.41i·20-s + (1.00 + 1.00i)22-s − 1.00·25-s − 1.41i·26-s − 1.41·32-s + 1.41·41-s + (0.707 + 0.707i)44-s − 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.120981772\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120981772\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 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
−10.13147666529101473514834987981, −9.353890148339358385011275156057, −8.085145865242392571968158165167, −7.10089717010143671546310084791, −6.55071405959281901704079285525, −5.76913589984929420473186488260, −4.82657331814778032184019446496, −3.80099604081656560448266874899, −3.13782929097991862396814459764, −2.19849562574212722740814890715,
1.39101788316382092332567372647, 2.85586617580369934813398021190, 4.15145549029434187090176070890, 4.40550391454081969945127358542, 5.47653976344901520617722588143, 6.07281777113144528444814171380, 7.00790473567206659989957567947, 8.230157115431977138748214605092, 9.057539118186760601728593504993, 9.438063365195954507601409502679