L(s) = 1 | + (1.25 − 2.17i)2-s + (−0.5 + 0.866i)3-s + (−2.16 − 3.74i)4-s + (−1.66 + 2.87i)5-s + (1.25 + 2.17i)6-s + 2.32·7-s − 5.83·8-s + (−0.499 − 0.866i)9-s + (4.17 + 7.23i)10-s − 1.70·11-s + 4.32·12-s + (−2.01 − 3.48i)13-s + (2.91 − 5.05i)14-s + (−1.66 − 2.87i)15-s + (−3.01 + 5.22i)16-s + ⋯ |
L(s) = 1 | + (0.888 − 1.53i)2-s + (−0.288 + 0.499i)3-s + (−1.08 − 1.87i)4-s + (−0.742 + 1.28i)5-s + (0.513 + 0.888i)6-s + 0.877·7-s − 2.06·8-s + (−0.166 − 0.288i)9-s + (1.32 + 2.28i)10-s − 0.514·11-s + 1.24·12-s + (−0.558 − 0.967i)13-s + (0.779 − 1.35i)14-s + (−0.428 − 0.742i)15-s + (−0.753 + 1.30i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.846487 - 0.652629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.846487 - 0.652629i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.193 - 4.35i)T \) |
good | 2 | \( 1 + (-1.25 + 2.17i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.66 - 2.87i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + (2.01 + 3.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 2.03i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.32 + 5.75i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-3.32 + 5.75i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.353 - 0.612i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.98 - 8.62i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.853 - 1.47i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.69 + 2.93i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.18 - 7.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.70 + 8.15i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.82 - 10.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.67 + 2.90i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + (-1.33 - 2.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.86 - 15.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.81197389192486419983691683028, −13.87997974762801974695047236037, −12.39486437786918420430776271376, −11.47038008537841220191263069048, −10.74316328484034785040043227968, −9.981722622473345269012849564950, −7.77450293761169913236384536454, −5.57222427665717340383168612318, −4.14833953162062799113937923518, −2.81871623103593173270887540885,
4.51778149694231630428310060811, 5.17781613996977954792779011068, 6.90563790575456069500084841087, 7.957880378968913562745530172828, 8.818723326978585894456721069922, 11.51545091123422372035104233252, 12.49214142205402292838886570007, 13.35633797664994695896872278126, 14.46507060263240882713199873838, 15.51868418320865219810418344004