L(s) = 1 | + (−1.22 − 0.707i)3-s + (−0.5 + 0.866i)5-s + 7-s + (0.499 + 0.866i)9-s − 11-s + (1.22 − 0.707i)15-s + (−0.5 + 0.866i)17-s + (−1.22 − 0.707i)21-s + (−1.22 + 0.707i)29-s + (1.22 + 0.707i)33-s + (−0.5 + 0.866i)35-s + 1.41i·37-s + (1.22 + 0.707i)41-s + (0.5 − 0.866i)43-s − 0.999·45-s + ⋯ |
L(s) = 1 | + (−1.22 − 0.707i)3-s + (−0.5 + 0.866i)5-s + 7-s + (0.499 + 0.866i)9-s − 11-s + (1.22 − 0.707i)15-s + (−0.5 + 0.866i)17-s + (−1.22 − 0.707i)21-s + (−1.22 + 0.707i)29-s + (1.22 + 0.707i)33-s + (−0.5 + 0.866i)35-s + 1.41i·37-s + (1.22 + 0.707i)41-s + (0.5 − 0.866i)43-s − 0.999·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5199397366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5199397366\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−10.37094212318765245676029884050, −8.951061328634858043950061826256, −7.931689125367679332791471999412, −7.42727232741424917232227220032, −6.66921148377182522444344660130, −5.78798385778180129317727102002, −5.10413012117293280495310975539, −4.05663252977673038682526547770, −2.70395402926573974486202303440, −1.44508430544624958571218159110,
0.51779114806239888924005940377, 2.27662760487387646949422837312, 4.00691161571004846615683528141, 4.66640786259323353885252842200, 5.27328700807958000142946518295, 5.88395725373288879281953477885, 7.25527833936559984764206287036, 7.940154236470389709550281088689, 8.800606153990438701649300695698, 9.637506105781492931175894743496