L(s) = 1 | + (−1 + 1.73i)5-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s − 2·17-s + 5·19-s + (−1 + 1.73i)23-s + (0.500 + 0.866i)25-s + (−2.5 + 4.33i)31-s − 6·37-s + (−6 + 10.3i)41-s + (2.5 + 4.33i)43-s + (−6 − 10.3i)47-s + (3.5 − 6.06i)49-s − 10·53-s + 3.99·55-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (−0.301 − 0.522i)11-s + (−0.138 + 0.240i)13-s − 0.485·17-s + 1.14·19-s + (−0.208 + 0.361i)23-s + (0.100 + 0.173i)25-s + (−0.449 + 0.777i)31-s − 0.986·37-s + (−0.937 + 1.62i)41-s + (0.381 + 0.660i)43-s + (−0.875 − 1.51i)47-s + (0.5 − 0.866i)49-s − 1.37·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5622397116\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5622397116\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.555781631288994323306813265795, −8.717112458364924271867745259311, −7.901127986215215897550834075025, −7.16755265325686502774135531847, −6.53639345570329810052441702556, −5.51808089034684217524515146102, −4.70308990383988135192720322330, −3.47640319368073363095881930124, −2.99272767887030108309912909637, −1.57854085616787546879004171627,
0.20066866674333013713527648214, 1.62583943622135293654993622000, 2.86320527051923151175227086061, 3.96734418509576814912329639412, 4.78995843961777833227022980266, 5.45091547667900021233988424707, 6.50573620683339847625912197886, 7.46483011733417747709095168166, 7.981889112328481606351953630916, 8.906764748174508435648924970428