L(s) = 1 | + 5.61i·3-s + 7.03·7-s − 22.5·9-s − 1.02·11-s − 11.3i·13-s − 16.5·17-s + (9.54 + 16.4i)19-s + 39.4i·21-s − 35.1·23-s − 75.9i·27-s + 6.63i·29-s − 25.3i·31-s − 5.73i·33-s − 47.8i·37-s + 63.7·39-s + ⋯ |
L(s) = 1 | + 1.87i·3-s + 1.00·7-s − 2.50·9-s − 0.0928·11-s − 0.872i·13-s − 0.975·17-s + (0.502 + 0.864i)19-s + 1.88i·21-s − 1.52·23-s − 2.81i·27-s + 0.228i·29-s − 0.818i·31-s − 0.173i·33-s − 1.29i·37-s + 1.63·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4070410187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4070410187\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-9.54 - 16.4i)T \) |
good | 3 | \( 1 - 5.61iT - 9T^{2} \) |
| 7 | \( 1 - 7.03T + 49T^{2} \) |
| 11 | \( 1 + 1.02T + 121T^{2} \) |
| 13 | \( 1 + 11.3iT - 169T^{2} \) |
| 17 | \( 1 + 16.5T + 289T^{2} \) |
| 23 | \( 1 + 35.1T + 529T^{2} \) |
| 29 | \( 1 - 6.63iT - 841T^{2} \) |
| 31 | \( 1 + 25.3iT - 961T^{2} \) |
| 37 | \( 1 + 47.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 78.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 45.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 9.44iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 45.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 144. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 56.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 81.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 103. iT - 9.40e3T^{2} \) |
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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
−9.105295404491284118115718496332, −8.166387963080107605506892832883, −7.74970298189430172832600072113, −6.09124989626095524678955155039, −5.52301712080862461594337353780, −4.67718596996736058008843095694, −4.08564417514219099796722456792, −3.21738834226985476781283267101, −2.04228749795482789647504606258, −0.10063367829220773898271280290,
1.26962603462727380252331800406, 1.96571668935278188111139857162, 2.82772344368122806498562725892, 4.34386346393553438333928924704, 5.27794117505034768807069655407, 6.27568973683257851554038605666, 6.83857777159052439784089476143, 7.54454207984562644819955220279, 8.305292005043650422394668657654, 8.721669380573608878538626059540