L(s) = 1 | + (0.980 − 0.195i)2-s + (0.923 − 0.382i)4-s + (−0.831 − 0.555i)7-s + (0.831 − 0.555i)8-s + (0.555 + 0.831i)9-s + (0.187 + 0.0569i)11-s + (−0.923 − 0.382i)14-s + (0.707 − 0.707i)16-s + (0.707 + 0.707i)18-s + (0.195 + 0.0192i)22-s + (−1.08 − 0.216i)23-s + (−0.195 − 0.980i)25-s + (−0.980 − 0.195i)28-s + (0.368 + 1.21i)29-s + (0.555 − 0.831i)32-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)2-s + (0.923 − 0.382i)4-s + (−0.831 − 0.555i)7-s + (0.831 − 0.555i)8-s + (0.555 + 0.831i)9-s + (0.187 + 0.0569i)11-s + (−0.923 − 0.382i)14-s + (0.707 − 0.707i)16-s + (0.707 + 0.707i)18-s + (0.195 + 0.0192i)22-s + (−1.08 − 0.216i)23-s + (−0.195 − 0.980i)25-s + (−0.980 − 0.195i)28-s + (0.368 + 1.21i)29-s + (0.555 − 0.831i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.722168838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722168838\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.980 + 0.195i)T \) |
| 7 | \( 1 + (0.831 + 0.555i)T \) |
good | 3 | \( 1 + (-0.555 - 0.831i)T^{2} \) |
| 5 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 11 | \( 1 + (-0.187 - 0.0569i)T + (0.831 + 0.555i)T^{2} \) |
| 13 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 19 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 23 | \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.368 - 1.21i)T + (-0.831 + 0.555i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (1.90 - 0.187i)T + (0.980 - 0.195i)T^{2} \) |
| 41 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (0.512 - 0.273i)T + (0.555 - 0.831i)T^{2} \) |
| 47 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 53 | \( 1 + (0.273 - 0.902i)T + (-0.831 - 0.555i)T^{2} \) |
| 59 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 61 | \( 1 + (0.555 + 0.831i)T^{2} \) |
| 67 | \( 1 + (0.831 - 1.55i)T + (-0.555 - 0.831i)T^{2} \) |
| 71 | \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (0.149 + 0.360i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 89 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.32461727892698143047227659335, −9.868235702749385174180211012106, −8.485933964522635995647594344201, −7.39051973448962196871834870225, −6.77006037287794697242910226139, −5.87562423335578325933120308628, −4.78883727244468229802982206696, −4.01093299028336436977281214138, −2.98916121807563719093372701726, −1.71186877923010154005933932169,
1.96008175230361759246814383679, 3.30990162859056515897050924786, 3.93220546484302176312141692145, 5.17581062460707438753543201310, 6.10465747575296515853461592180, 6.67879520069313952194432309196, 7.58862990942791973706012387515, 8.681008964982522384916826810520, 9.643957765897749253650774958967, 10.34780492915808710909685485167