L(s) = 1 | + (0.587 + 0.190i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (0.587 − 0.809i)5-s + (0.587 − 0.190i)6-s + (0.309 + 0.951i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.5 − 0.363i)10-s + (−0.951 + 1.30i)11-s − 0.618·12-s + (−0.309 − 0.951i)13-s + 0.618i·14-s − i·15-s + (−1.53 + 0.5i)17-s + (0.363 − 0.5i)18-s + ⋯ |
L(s) = 1 | + (0.587 + 0.190i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (0.587 − 0.809i)5-s + (0.587 − 0.190i)6-s + (0.309 + 0.951i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.5 − 0.363i)10-s + (−0.951 + 1.30i)11-s − 0.618·12-s + (−0.309 − 0.951i)13-s + 0.618i·14-s − i·15-s + (−1.53 + 0.5i)17-s + (0.363 − 0.5i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.424903184\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.424903184\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 211 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 11 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−10.42146647802318201931377819982, −9.623608358695255102892517647332, −8.920286659729656062732035206484, −8.241507329420535546343917990844, −7.14651444386403013070833566956, −5.95876125910742687971610051907, −5.19983729813939363211756870668, −4.39724917669075648191804250101, −2.81098382548455277816684619370, −1.68981350742257403663191874460,
2.57607739656722820028388863709, 3.10575332720359491674074358631, 4.44885057391912521562496132714, 4.90847086402005982810457730465, 6.43839664091665218565118665229, 7.42700049860312932566113912544, 8.473595037881118737030248496591, 9.093789861059388216814220409667, 10.08687028025800864930051571447, 11.00809876593834946538447205645