L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.448 + 1.67i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)23-s + (−1.22 + 1.22i)28-s + (−0.866 − 1.5i)29-s + (0.258 + 0.965i)32-s + (1.5 + 0.866i)41-s − 46-s + (0.965 + 0.258i)47-s + (−1.73 − 1.00i)49-s + (−1.5 + 0.866i)56-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.448 + 1.67i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)23-s + (−1.22 + 1.22i)28-s + (−0.866 − 1.5i)29-s + (0.258 + 0.965i)32-s + (1.5 + 0.866i)41-s − 46-s + (0.965 + 0.258i)47-s + (−1.73 − 1.00i)49-s + (−1.5 + 0.866i)56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.094845548\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.094845548\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T^{2} \) |
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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
−9.231956774161275317682420157360, −8.230544127569965244408593973986, −7.72354466538025876524737361997, −6.61549892593415280129313498329, −5.92565397600217609361934136045, −5.56490505899758628988315463265, −4.53146501544152355593297300270, −3.63768598044373356370222390827, −2.64844653282263716906193318906, −2.03152601770716731794644619584,
0.998476391890196858342082375831, 2.25078462110706572923472945314, 3.47922817753105643036856906429, 3.93865983345903767771622835380, 4.74430294758018351009397145535, 5.71727282666781441473073774330, 6.49830506702878249363359673043, 7.27645379368514398810762025116, 7.64997772898529685983397782333, 8.953684325383495857941957041438