L(s) = 1 | + (1 + 1.73i)5-s + (1 − 1.73i)7-s + (−1 + 1.73i)11-s + (0.5 + 0.866i)13-s − 6·17-s − 6·19-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s + (1 − 1.73i)29-s + (−5 − 8.66i)31-s + 3.99·35-s − 6·37-s + (−3 − 5.19i)41-s + (−2 + 3.46i)43-s + (−1 + 1.73i)47-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.377 − 0.654i)7-s + (−0.301 + 0.522i)11-s + (0.138 + 0.240i)13-s − 1.45·17-s − 1.37·19-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s + (0.185 − 0.321i)29-s + (−0.898 − 1.55i)31-s + 0.676·35-s − 0.986·37-s + (−0.468 − 0.811i)41-s + (−0.304 + 0.528i)43-s + (−0.145 + 0.252i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−7.935402086638341424766671292984, −7.29435072098495924530384693622, −6.63221975556018327223869234947, −6.07062420617015257892271551804, −4.96715169147961621429874312548, −4.32434264461534426660229144821, −3.46544225614783731464777974699, −2.33658408283403394655731361427, −1.73239042129283392178990796261, 0,
1.44848883123461178262717839400, 2.30304545456749017067089545834, 3.22416581993473666532195864540, 4.48793479799210210532315241885, 4.94846342074959796905846944028, 5.70292559592217599261584780939, 6.51295219091066507792724493691, 7.13960985732195713161698660566, 8.481251033826456252685701795446