L(s) = 1 | − 2.49·2-s + (−1.44 + 2.49i)3-s + 4.21·4-s + (0.778 + 1.34i)5-s + (3.59 − 6.22i)6-s + (−0.277 + 0.481i)7-s − 5.52·8-s + (−2.65 − 4.59i)9-s + (−1.94 − 3.36i)10-s + (−1.79 − 3.11i)11-s + (−6.07 + 10.5i)12-s + (−0.5 − 0.866i)13-s + (0.692 − 1.19i)14-s − 4.48·15-s + 5.34·16-s + (−1.01 + 1.75i)17-s + ⋯ |
L(s) = 1 | − 1.76·2-s + (−0.831 + 1.44i)3-s + 2.10·4-s + (0.348 + 0.603i)5-s + (1.46 − 2.53i)6-s + (−0.105 + 0.181i)7-s − 1.95·8-s + (−0.883 − 1.53i)9-s + (−0.613 − 1.06i)10-s + (−0.541 − 0.937i)11-s + (−1.75 + 3.03i)12-s + (−0.138 − 0.240i)13-s + (0.185 − 0.320i)14-s − 1.15·15-s + 1.33·16-s + (−0.245 + 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0583378 - 0.0449459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0583378 - 0.0449459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-2.10 + 5.15i)T \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + (1.44 - 2.49i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.778 - 1.34i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.277 - 0.481i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.79 + 3.11i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.01 - 1.75i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.578 - 1.00i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 37 | \( 1 + (-4.06 + 7.04i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.94 + 10.2i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.78 - 3.09i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + (-4.10 - 7.11i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.11 - 3.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 0.963T + 61T^{2} \) |
| 67 | \( 1 + (2.74 + 4.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.95 + 3.38i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.90 - 5.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.38 - 5.86i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.967 + 1.67i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−10.63264514195022299768653885538, −10.26244747762265033154618724842, −9.482994907772485091066913583867, −8.624852135678100389698426552448, −7.63270420459387619629634104454, −6.21059359346036814910573820090, −5.71591003947702412653631988457, −3.96813140823285851497359012062, −2.47276794857448868581971536384, −0.094497343625998001314373657846,
1.37034640457333119587453702330, 2.25842550537183813263395159371, 5.04305281985333095388076560472, 6.30207564849889292241543856242, 7.01300846244492649715135451137, 7.76134227638552169144350477924, 8.541844011405943239484311190628, 9.686694349020406356052718806564, 10.33956695158355293298296945862, 11.50092674482898475642807879579