L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + i·5-s − 1.00·6-s − 7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (0.707 + 0.707i)15-s − 1.00·16-s + i·17-s + (0.707 + 0.707i)19-s − 1.00·20-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + i·5-s − 1.00·6-s − 7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (0.707 + 0.707i)15-s − 1.00·16-s + i·17-s + (0.707 + 0.707i)19-s − 1.00·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7835125603\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7835125603\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 - 2iT - T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.566926179539050697063234793515, −9.147114225966245265821650614149, −8.019463094750664397749578840028, −7.44158713749325218652908747655, −6.88074002683771779680644685180, −5.94036302323436948812801378120, −4.22291508916034734299642246034, −3.24449585283641692459417404995, −2.62024498961102434629209962522, −1.66824405569572597715882493225,
0.73608104669461983488924517008, 2.58849494544677739444930765225, 3.61128416579616861925247474521, 4.91095372551600153974794357312, 5.31502011100506187701412979596, 6.55685334947710657358360249589, 7.32261604590078235452069942766, 8.184844214934337419255514428214, 9.047070660660268335780254105304, 9.460674522501369346209944625092