L(s) = 1 | + (0.707 + 0.707i)3-s + (0.866 − 0.5i)4-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (−1.5 − 0.866i)11-s + (0.965 + 0.258i)12-s + (0.258 − 0.965i)13-s + (0.499 − 0.866i)16-s + (1.22 + 1.22i)17-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s + (−0.448 − 1.67i)33-s + (0.500 + 0.866i)36-s + (0.866 − 0.500i)39-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (0.866 − 0.5i)4-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (−1.5 − 0.866i)11-s + (0.965 + 0.258i)12-s + (0.258 − 0.965i)13-s + (0.499 − 0.866i)16-s + (1.22 + 1.22i)17-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s + (−0.448 − 1.67i)33-s + (0.500 + 0.866i)36-s + (0.866 − 0.500i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.667492659\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.667492659\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.258 - 0.965i)T \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.879468931863115618844300433755, −8.813257866094117916965841286300, −7.957013807370502891355802428441, −7.79870369056022908507108860424, −6.13303317335858496087619204206, −5.59785560801889948666427073498, −4.95755143725239335877552336851, −3.31253900829845018435878211664, −2.88155032220763718206105915475, −1.75980014867132028017009600836,
1.46692694034805235422491292652, 2.49056154912144387516855661431, 3.29643622920929884345041755892, 4.35882998953907408833590250625, 5.51615843865987847402433389386, 6.84624108275772882350129648185, 7.14604109233466217091744598685, 7.83752397576515430740218704552, 8.385444913631171517676825371695, 9.672703345872948066689674243341