L(s) = 1 | − 2.00i·3-s + (4.99 − 0.247i)5-s + 4.85·7-s + 4.98·9-s + 0.0522·11-s − 8.88·13-s + (−0.496 − 10.0i)15-s + 3.74i·17-s + 20.3·19-s − 9.73i·21-s + 15.3·23-s + (24.8 − 2.47i)25-s − 28.0i·27-s + 38.3i·29-s − 34.5i·31-s + ⋯ |
L(s) = 1 | − 0.668i·3-s + (0.998 − 0.0495i)5-s + 0.693·7-s + 0.553·9-s + 0.00475·11-s − 0.683·13-s + (−0.0331 − 0.667i)15-s + 0.220i·17-s + 1.07·19-s − 0.463i·21-s + 0.668·23-s + (0.995 − 0.0990i)25-s − 1.03i·27-s + 1.32i·29-s − 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.895302604\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.895302604\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.99 + 0.247i)T \) |
good | 3 | \( 1 + 2.00iT - 9T^{2} \) |
| 7 | \( 1 - 4.85T + 49T^{2} \) |
| 11 | \( 1 - 0.0522T + 121T^{2} \) |
| 13 | \( 1 + 8.88T + 169T^{2} \) |
| 17 | \( 1 - 3.74iT - 289T^{2} \) |
| 19 | \( 1 - 20.3T + 361T^{2} \) |
| 23 | \( 1 - 15.3T + 529T^{2} \) |
| 29 | \( 1 - 38.3iT - 841T^{2} \) |
| 31 | \( 1 + 34.5iT - 961T^{2} \) |
| 37 | \( 1 + 13.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 5.41iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.15T + 2.80e3T^{2} \) |
| 59 | \( 1 + 88.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 90.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 71.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 113. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 1.80iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 135. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 70.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 106. iT - 9.40e3T^{2} \) |
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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
−9.411245973505705589883629574455, −8.608615909656190649036587571984, −7.49050351345034319013779959682, −7.12676906307643548082341916831, −6.03391313216766464102196733894, −5.25118529053667575294517926063, −4.39515651552108686113390345425, −2.89622409980574300437477119333, −1.86574235230660007096561003861, −1.04098830681143824804967764772,
1.16269082474543862665225059387, 2.29562421304024187612666450963, 3.43715856400615967925576373579, 4.75727159576150170353303701877, 5.07439024054068103562405254245, 6.16406153730252672772712524936, 7.16616052995717517831372850213, 7.891690019541504119096362132428, 9.191526249871295887402721993132, 9.447635927538375622262376783470