L(s) = 1 | + (3.47 − 2.00i)5-s + (2.66 + 1.54i)7-s + (−2.97 + 5.15i)11-s + (0.673 + 1.16i)13-s − 4.87i·17-s − 7.49i·19-s + (1.28 + 2.23i)23-s + (5.57 − 9.65i)25-s + (5.60 + 3.23i)29-s + (−1.28 + 0.743i)31-s + 12.3·35-s + 1.91·37-s + (−1.69 + 0.978i)41-s + (6.79 + 3.92i)43-s + (1.14 − 1.97i)47-s + ⋯ |
L(s) = 1 | + (1.55 − 0.898i)5-s + (1.00 + 0.582i)7-s + (−0.897 + 1.55i)11-s + (0.186 + 0.323i)13-s − 1.18i·17-s − 1.71i·19-s + (0.268 + 0.465i)23-s + (1.11 − 1.93i)25-s + (1.04 + 0.600i)29-s + (−0.231 + 0.133i)31-s + 2.09·35-s + 0.315·37-s + (−0.264 + 0.152i)41-s + (1.03 + 0.598i)43-s + (0.166 − 0.288i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.770476113\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.770476113\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.47 + 2.00i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.66 - 1.54i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.97 - 5.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.673 - 1.16i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.87iT - 17T^{2} \) |
| 19 | \( 1 + 7.49iT - 19T^{2} \) |
| 23 | \( 1 + (-1.28 - 2.23i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.60 - 3.23i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.28 - 0.743i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 + (1.69 - 0.978i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.79 - 3.92i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.14 + 1.97i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.871iT - 53T^{2} \) |
| 59 | \( 1 + (-0.650 - 1.12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.29 + 5.71i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.44 + 4.87i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.64T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 + (-8.08 - 4.66i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.09 - 8.82i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 0.947iT - 89T^{2} \) |
| 97 | \( 1 + (5.15 - 8.93i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.085381409867922853301874569718, −8.244291868700137073191680521889, −7.29078069238096402890211867913, −6.57584538604306492224265071084, −5.42121507826044351858444775455, −4.95015399121056364998657831060, −4.64523029379681152817901236265, −2.61554687001850437465123287087, −2.16781810059198202866422807815, −1.12017346843600912816695238848,
1.15532499253498148061827781296, 2.14293528693536373319296546910, 3.05600867669114899154856781891, 4.01174508546998875413823340621, 5.27873887561089373554989174143, 5.91668651612375319273872713909, 6.28710878681731137460553417102, 7.47555450522355243940387980816, 8.183076796415592722928778985558, 8.731320988289535918566784919697