L(s) = 1 | + (−0.302 − 3.98i)2-s − 5.19i·3-s + (−15.8 + 2.41i)4-s + 34.8·5-s + (−20.7 + 1.57i)6-s + 43.0i·7-s + (14.4 + 62.3i)8-s − 27·9-s + (−10.5 − 138. i)10-s − 65.2i·11-s + (12.5 + 82.1i)12-s − 99.0·13-s + (171. − 13.0i)14-s − 181. i·15-s + (244. − 76.4i)16-s − 207.·17-s + ⋯ |
L(s) = 1 | + (−0.0756 − 0.997i)2-s − 0.577i·3-s + (−0.988 + 0.150i)4-s + 1.39·5-s + (−0.575 + 0.0437i)6-s + 0.878i·7-s + (0.225 + 0.974i)8-s − 0.333·9-s + (−0.105 − 1.38i)10-s − 0.539i·11-s + (0.0871 + 0.570i)12-s − 0.586·13-s + (0.875 − 0.0664i)14-s − 0.804i·15-s + (0.954 − 0.298i)16-s − 0.718·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.835283 - 0.717415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.835283 - 0.717415i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.302 + 3.98i)T \) |
| 3 | \( 1 + 5.19iT \) |
good | 5 | \( 1 - 34.8T + 625T^{2} \) |
| 7 | \( 1 - 43.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 65.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 99.0T + 2.85e4T^{2} \) |
| 17 | \( 1 + 207.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 569. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 371. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 423.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.17e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.44e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 265.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 699. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 787.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.01e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.51e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.21e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.69e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.40e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.45e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.95e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 616.T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.37e4T + 8.85e7T^{2} \) |
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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
−19.01412484068610789336533989452, −18.15794452648903551197687389496, −17.02558031384986871131911013558, −14.39756887227521205062159558897, −13.24138420205178385669881740226, −12.00448445958351114897360335675, −10.16515386371985889655554108940, −8.714537531290301440109738135882, −5.74892451087126920716738699033, −2.19838578255000481774688793477,
4.94391572785138053514189951270, 6.86371007993304063065754336602, 9.191546723912840722951423572117, 10.31970931103209328325156031806, 13.26630440922724312702570987254, 14.25169970326605343776020203273, 15.71325364347190545992028818705, 17.28972121736108648154944507251, 17.65561086366146165583328136514, 19.80409457020598747465178587368