L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − i·9-s + 11-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (0.965 − 0.258i)18-s + (0.258 + 0.965i)22-s + (−0.707 − 0.707i)23-s + (−0.258 + 0.965i)28-s + 1.73·29-s + (0.965 + 0.258i)32-s + (0.499 + 0.866i)36-s + (−0.707 + 0.707i)37-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − i·9-s + 11-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (0.965 − 0.258i)18-s + (0.258 + 0.965i)22-s + (−0.707 − 0.707i)23-s + (−0.258 + 0.965i)28-s + 1.73·29-s + (0.965 + 0.258i)32-s + (0.499 + 0.866i)36-s + (−0.707 + 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.241942799\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.241942799\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 71 | \( 1 - 1.73iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.731536676338040947745420589062, −8.721773179583934807554885694132, −8.302809230561816843189581601759, −7.21290953208456730717277388383, −6.61916048836968138814970830174, −5.92967132053309118549805858668, −4.66813132223583485105189495716, −4.16095688480089205738650586775, −3.14911021694096398364553374199, −1.15519956538534857666304178869,
1.56004019184598016404971981003, 2.35929393547405430867988863662, 3.57301485632991461757942915631, 4.57569577700303725254945155752, 5.27045236779371516466546121510, 6.11065440767563083960851607914, 7.35049577169013945838907484702, 8.517714463964536355190350963375, 8.738063755086060348955316149548, 9.921650215404377047217249030838