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
−10.34780492915808710909685485167, −9.643957765897749253650774958967, −8.681008964982522384916826810520, −7.58862990942791973706012387515, −6.67879520069313952194432309196, −6.10465747575296515853461592180, −5.17581062460707438753543201310, −3.93220546484302176312141692145, −3.30990162859056515897050924786, −1.96008175230361759246814383679,
1.71186877923010154005933932169, 2.98916121807563719093372701726, 4.01093299028336436977281214138, 4.78883727244468229802982206696, 5.87562423335578325933120308628, 6.77006037287794697242910226139, 7.39051973448962196871834870225, 8.485933964522635995647594344201, 9.868235702749385174180211012106, 10.32461727892698143047227659335