| L(s) = 1 | + (−1.44 − 0.907i)2-s + (−0.472 − 0.980i)4-s + (8.15 − 1.86i)5-s + (7.53 + 3.62i)7-s + (−0.972 + 8.62i)8-s + (−13.4 − 4.71i)10-s + (1.08 + 9.61i)11-s + (8.88 + 7.08i)13-s + (−7.59 − 12.0i)14-s + (6.52 − 8.17i)16-s + (−11.2 + 11.2i)17-s + (−5.31 + 15.1i)19-s + (−5.67 − 7.12i)20-s + (7.16 − 14.8i)22-s + (2.98 − 13.0i)23-s + ⋯ |
| L(s) = 1 | + (−0.722 − 0.453i)2-s + (−0.118 − 0.245i)4-s + (1.63 − 0.372i)5-s + (1.07 + 0.518i)7-s + (−0.121 + 1.07i)8-s + (−1.34 − 0.471i)10-s + (0.0985 + 0.874i)11-s + (0.683 + 0.544i)13-s + (−0.542 − 0.863i)14-s + (0.407 − 0.511i)16-s + (−0.664 + 0.664i)17-s + (−0.279 + 0.799i)19-s + (−0.283 − 0.356i)20-s + (0.325 − 0.676i)22-s + (0.129 − 0.569i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48641 - 0.274547i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48641 - 0.274547i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (-25.9 - 13.0i)T \) |
| good | 2 | \( 1 + (1.44 + 0.907i)T + (1.73 + 3.60i)T^{2} \) |
| 5 | \( 1 + (-8.15 + 1.86i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 + (-7.53 - 3.62i)T + (30.5 + 38.3i)T^{2} \) |
| 11 | \( 1 + (-1.08 - 9.61i)T + (-117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-8.88 - 7.08i)T + (37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (11.2 - 11.2i)T - 289iT^{2} \) |
| 19 | \( 1 + (5.31 - 15.1i)T + (-282. - 225. i)T^{2} \) |
| 23 | \( 1 + (-2.98 + 13.0i)T + (-476. - 229. i)T^{2} \) |
| 31 | \( 1 + (-1.32 - 0.832i)T + (416. + 865. i)T^{2} \) |
| 37 | \( 1 + (-2.36 + 21.0i)T + (-1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (-0.193 - 0.193i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (38.8 + 61.7i)T + (-802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (42.6 - 4.80i)T + (2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (17.1 + 75.1i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 - 23.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-56.8 + 19.9i)T + (2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (48.7 - 38.9i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (79.7 + 63.6i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (23.3 - 14.6i)T + (2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 1.32i)T + (6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (0.116 - 0.0561i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (24.8 + 15.6i)T + (3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (-23.5 - 8.24i)T + (7.35e3 + 5.86e3i)T^{2} \) |
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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
−11.47029443065696212001096045086, −10.47715569142436932122781154812, −9.874548687469442589890990426798, −8.828364714475051633750450986651, −8.425439344827548566374898322109, −6.54126824610824125897395987189, −5.53815999033473641812127515666, −4.65726151382853360983168098620, −2.06464083273635425053291212946, −1.61198215555362390406837778706,
1.18717319806482994445919381717, 2.95140409096722513258704716444, 4.69193318107794937463130251729, 6.01519207770912580533714641548, 6.87423373638097111186208797243, 8.070277745288912829228309429586, 8.860694211558930999181561291664, 9.763067555207455727454345260742, 10.67725413246480287172352067439, 11.50488370839278211492536077917
