L(s) = 1 | + (0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.984 − 0.173i)13-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)23-s + (0.342 + 0.939i)29-s + (−0.5 + 0.866i)31-s + i·37-s + (−0.173 − 0.984i)41-s + (0.642 + 0.766i)43-s + (0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + (0.642 − 0.766i)53-s + (−0.342 + 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.984 − 0.173i)13-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)23-s + (0.342 + 0.939i)29-s + (−0.5 + 0.866i)31-s + i·37-s + (−0.173 − 0.984i)41-s + (0.642 + 0.766i)43-s + (0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + (0.642 − 0.766i)53-s + (−0.342 + 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8907745726 + 0.8331727776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8907745726 + 0.8331727776i\) |
\(L(1)\) |
\(\approx\) |
\(1.033512866 - 0.07748038667i\) |
\(L(1)\) |
\(\approx\) |
\(1.033512866 - 0.07748038667i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.342 + 0.939i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.642 - 0.766i)T \) |
| 59 | \( 1 + (-0.342 + 0.939i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (-0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.72350583596957879395220966329, −17.34597363635816430424803233008, −16.700689101593718801227852330376, −15.7610444390611379987125861279, −15.05041697623442306021445582616, −14.73121970259052668831294876946, −13.99706814561192708183815633353, −13.05654807261770342292089488594, −12.40891861937943730708893677750, −11.84474483869933604275209129150, −11.21146242754414714135835502633, −10.45036242403134583694394419747, −9.45219368829076981403393031221, −9.11751473273062461571104570207, −8.341282362789418016240750965740, −7.49895934933479696591962230699, −6.81887833337584268874907795001, −6.05609944130786316162810545293, −5.29868631579294566492592924573, −4.44652033792722030549570799393, −4.01804615980297626314372764979, −2.59601894620824731854719050940, −2.293539218169637900649540957294, −1.32700635765032562158363193879, −0.1946352723243309109593842927,
0.83797405361439387667084323454, 1.50944106345643861195030223308, 2.53335776029986447477750861683, 3.411193140292724911714831504026, 4.14488045058156425899340936684, 4.91442914346603576990493062988, 5.49544502153155576759170198403, 6.83668694038013793905815319226, 6.89632043227849560086425615999, 7.84310163206466385199587039824, 8.679970480195067775843112705078, 9.2622976881421509391973203509, 10.07648839489617728058645044363, 10.864313815829200782455746898793, 11.30424094667595463014018390472, 12.12036698590100299664143176440, 12.801416927457434063866052609501, 13.74465852275378196388673019383, 14.04543648377433894977700376278, 14.84089078540185837311637753782, 15.44429846139471625842308644024, 16.39917198128787667487510943749, 16.91127649319413725472120638206, 17.51511631695906974088507517388, 18.03447899602978586405086021433