| L(s) = 1 | + 135. i·3-s − 581.·5-s + 2.67e3i·7-s − 1.19e4·9-s − 1.41e4i·11-s + 4.28e4·13-s − 7.90e4i·15-s − 1.40e5·17-s − 1.29e5i·19-s − 3.62e5·21-s + 1.96e5i·23-s − 5.22e4·25-s − 7.27e5i·27-s − 3.55e5·29-s − 3.35e3i·31-s + ⋯ |
| L(s) = 1 | + 1.67i·3-s − 0.930·5-s + 1.11i·7-s − 1.81·9-s − 0.966i·11-s + 1.49·13-s − 1.56i·15-s − 1.68·17-s − 0.995i·19-s − 1.86·21-s + 0.703i·23-s − 0.133·25-s − 1.36i·27-s − 0.503·29-s − 0.00363i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.229040 - 0.552952i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.229040 - 0.552952i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 135. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 581.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 2.67e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.41e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.28e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.40e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.29e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.96e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.55e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 3.35e3iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 9.07e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.06e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.01e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 3.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.34e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 7.78e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.42e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 6.87e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 5.76e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.19e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.55e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.03e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.19e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 3.19e7T + 7.83e15T^{2} \) |
| 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
−15.57206695491111864912712524943, −15.24912527072710623246610824620, −13.48297849847818767015006704964, −11.42694258607022845690115440073, −11.04686666148097301715199130532, −9.180300247733256859018160395106, −8.502570113137408173272631281172, −5.93537970653782596092755562246, −4.41460824133692604749422474687, −3.18369814818584980981364185652,
0.25328177459807282278534473988, 1.72129451089016326807996198709, 3.98413711600608749817441039965, 6.49194101977894172402773192351, 7.41715028560302545998238239567, 8.462325418541455206241791374352, 10.79311460963032039208021045444, 11.93824628558070040382219895465, 13.06854448663756853344112324211, 13.85187390250110299469149861654
