| L(s) = 1 | + 1.36i·5-s − 1.24i·7-s − 5.79·11-s + 16.3i·13-s + 5.01·17-s − 26.1·19-s + 25.1i·23-s + 23.1·25-s − 32.7i·29-s − 1.01i·31-s + 1.69·35-s − 14.9i·37-s − 72.5·41-s + 33.4·43-s − 66.5i·47-s + ⋯ |
| L(s) = 1 | + 0.272i·5-s − 0.177i·7-s − 0.527·11-s + 1.26i·13-s + 0.294·17-s − 1.37·19-s + 1.09i·23-s + 0.925·25-s − 1.13i·29-s − 0.0328i·31-s + 0.0484·35-s − 0.405i·37-s − 1.76·41-s + 0.778·43-s − 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3481833498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3481833498\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.36iT - 25T^{2} \) |
| 7 | \( 1 + 1.24iT - 49T^{2} \) |
| 11 | \( 1 + 5.79T + 121T^{2} \) |
| 13 | \( 1 - 16.3iT - 169T^{2} \) |
| 17 | \( 1 - 5.01T + 289T^{2} \) |
| 19 | \( 1 + 26.1T + 361T^{2} \) |
| 23 | \( 1 - 25.1iT - 529T^{2} \) |
| 29 | \( 1 + 32.7iT - 841T^{2} \) |
| 31 | \( 1 + 1.01iT - 961T^{2} \) |
| 37 | \( 1 + 14.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 33.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 20.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.641040515322010188562024589332, −7.69119724573228078719805389124, −6.98796633714866658698848492921, −6.29496452966241910995735605172, −5.38045182719581677490613960034, −4.43444603919222453450663804290, −3.69676263717660006867180655236, −2.54714638873836532051339456506, −1.64447603055918544278546263487, −0.087036586651657377322711075310,
1.14057533977969486875569051495, 2.48525119195016039860894019515, 3.23064931143003346948342523482, 4.44135693800747000081710280250, 5.13103068295654078813111346412, 5.95307298259323524310931962476, 6.76940998962718983296275030940, 7.67968090348424697738760979541, 8.468727164092733516834848212576, 8.852445572344077266036712544321
