L(s) = 1 | + 3-s − 5-s + (−0.5 − 0.866i)7-s + 9-s + (2.5 + 4.33i)11-s + (0.5 − 0.866i)13-s − 15-s + (3.5 − 6.06i)17-s + (2.5 − 4.33i)19-s + (−0.5 − 0.866i)21-s + (−4.5 + 7.79i)23-s + 25-s + 27-s + (1.5 + 2.59i)29-s + (3.5 + 6.06i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + (−0.188 − 0.327i)7-s + 0.333·9-s + (0.753 + 1.30i)11-s + (0.138 − 0.240i)13-s − 0.258·15-s + (0.848 − 1.47i)17-s + (0.573 − 0.993i)19-s + (−0.109 − 0.188i)21-s + (−0.938 + 1.62i)23-s + 0.200·25-s + 0.192·27-s + (0.278 + 0.482i)29-s + (0.628 + 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.313195828\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.313195828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-8 - 1.73i)T \) |
good | 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (3.5 + 6.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.5 - 14.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.449243014539680520881943608245, −7.62008891986542317429092759355, −7.10408990847386886559304565138, −6.61439409289064639552061258734, −5.16471241346136391153361672735, −4.80835440488526404172458958379, −3.61604525469188539450665387974, −3.23087212954232791513214625792, −2.00290762937205632472980029859, −0.936755714359830478958423710790,
0.816133910723787335645094288906, 1.97904142124449571270731278573, 3.06242961692403557541837558041, 3.83210798007507995258168335328, 4.26961363206489135727693717730, 5.83617337713193757661046648480, 5.98319911167567829114823617389, 6.98794517650558388553135528434, 7.999913497297040887747273978302, 8.382831543590931527937299037393