| L(s) = 1 | − 5.19·3-s − 39.7i·5-s − 46.0i·7-s + 27·9-s + 181.·11-s + 183. i·13-s + 206. i·15-s + 427.·17-s + 668.·19-s + 239. i·21-s − 882. i·23-s − 954.·25-s − 140.·27-s + 807. i·29-s + 391. i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.58i·5-s − 0.939i·7-s + 0.333·9-s + 1.49·11-s + 1.08i·13-s + 0.917i·15-s + 1.48·17-s + 1.85·19-s + 0.542i·21-s − 1.66i·23-s − 1.52·25-s − 0.192·27-s + 0.959i·29-s + 0.407i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.049814070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049814070\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19T \) |
| good | 5 | \( 1 + 39.7iT - 625T^{2} \) |
| 7 | \( 1 + 46.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 181.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 183. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 427.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 668.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 882. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 807. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 391. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 466. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.15e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 509.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 753. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.30e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 801. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 505.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.17e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.29e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.07e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 2.97e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.49e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 8.18e3T + 8.85e7T^{2} \) |
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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
−10.42735794466010323935854964207, −9.454323604706439702099807505681, −8.858993127269897940790682629602, −7.60215544316388559609731557567, −6.68529164529693433611837754275, −5.48018539835293951954286436778, −4.56296629385190070014944709783, −3.74909956355274498553773573873, −1.31084055014750739853321333572, −0.858276197064239112928949940639,
1.20375361168000848898650642868, 2.88154398004395523842857332563, 3.65602649165102490158983450169, 5.54981397638068052138227244556, 5.98074844966565183456382983609, 7.17377935453980964411318791171, 7.85222673920473517295738936313, 9.546164602638301024202619441614, 9.889130521830551157104063396767, 11.19034783694374386289933824499
