L(s) = 1 | + (−0.366 + 0.366i)5-s + (−0.866 − 0.5i)13-s + (−1.5 + 0.866i)17-s + 0.732i·25-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.133i)37-s + (0.5 + 1.86i)41-s + (0.866 + 0.5i)49-s − 1.73·53-s + (−0.5 − 0.866i)61-s + (0.5 − 0.133i)65-s + (−0.366 − 0.366i)73-s + (0.232 − 0.866i)85-s + (−1.36 + 0.366i)89-s + (1.36 + 0.366i)97-s + ⋯ |
L(s) = 1 | + (−0.366 + 0.366i)5-s + (−0.866 − 0.5i)13-s + (−1.5 + 0.866i)17-s + 0.732i·25-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.133i)37-s + (0.5 + 1.86i)41-s + (0.866 + 0.5i)49-s − 1.73·53-s + (−0.5 − 0.866i)61-s + (0.5 − 0.133i)65-s + (−0.366 − 0.366i)73-s + (0.232 − 0.866i)85-s + (−1.36 + 0.366i)89-s + (1.36 + 0.366i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5451760554\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5451760554\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 7 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + 1.73T + T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.000555995460260943396256960220, −8.118367822036802416857697135949, −7.53301772055677269152031115078, −6.77070027555303649423704147599, −6.11671430373085885341729484410, −5.11283148023672263099272972332, −4.43275356600897654752548610429, −3.48863486579827925638275280541, −2.67632431777107736698824264420, −1.60512409459421555681250842975,
0.29244452085144354466148667710, 1.97193332566613469593295784675, 2.71500682773321181389534093393, 4.00577669416578702898778452006, 4.52416585693641884165444578135, 5.30634038482444675740993940029, 6.26163185510634858362528600701, 7.07667809457908296256118972540, 7.56539721130351316671032109767, 8.552967682025295553220710667839