L(s) = 1 | + i·3-s + (0.489 + 2.18i)5-s + 2.82i·7-s − 9-s − 2.97·11-s − i·13-s + (−2.18 + 0.489i)15-s − 1.44i·17-s − 4.17·19-s − 2.82·21-s + 6.21i·23-s + (−4.52 + 2.13i)25-s − i·27-s + 0.828·29-s + 1.65·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.218 + 0.975i)5-s + 1.06i·7-s − 0.333·9-s − 0.898·11-s − 0.277i·13-s + (−0.563 + 0.126i)15-s − 0.350i·17-s − 0.956·19-s − 0.617·21-s + 1.29i·23-s + (−0.904 + 0.427i)25-s − 0.192i·27-s + 0.153·29-s + 0.297·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8823770038\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8823770038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.489 - 2.18i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 17 | \( 1 + 1.44iT - 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 6.21iT - 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 0.0418iT - 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 + 8.99iT - 43T^{2} \) |
| 47 | \( 1 - 4.40iT - 47T^{2} \) |
| 53 | \( 1 + 6.72iT - 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 + 0.727iT - 67T^{2} \) |
| 71 | \( 1 + 8.32T + 71T^{2} \) |
| 73 | \( 1 + 6.87iT - 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 6.78iT - 97T^{2} \) |
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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
−9.983544238742181955562336404790, −9.127770697041665496707415087174, −8.354393127973993433661746857960, −7.49965213362082306999576865106, −6.54096419665004333296573981415, −5.65851981098042855512567642756, −5.12466758967972978348186009545, −3.80394162690211273253729243355, −2.87126455673433378858816662331, −2.13898418732956772882879620457,
0.32795254603520698790392172039, 1.52570592846523494105127046913, 2.67335793089738469421783460881, 4.12110567199663120261721839597, 4.72620960278122994278930763150, 5.79161889354010319351125951481, 6.61871099634305029664909615412, 7.45537558029780828748033550904, 8.309106573704410257395299900194, 8.721412693735118153290887169638