L(s) = 1 | + (−0.923 + 0.382i)3-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.382 + 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s − 31-s + 33-s + (−0.382 + 0.923i)37-s + (−0.707 − 0.707i)39-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)3-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.382 + 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s − 31-s + 33-s + (−0.382 + 0.923i)37-s + (−0.707 − 0.707i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2389457912 + 0.3986569682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2389457912 + 0.3986569682i\) |
\(L(1)\) |
\(\approx\) |
\(0.6114101318 + 0.1392262160i\) |
\(L(1)\) |
\(\approx\) |
\(0.6114101318 + 0.1392262160i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.923 + 0.382i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + (0.382 + 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.382 + 0.923i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
| 29 | \( 1 + (-0.923 + 0.382i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.382 + 0.923i)T \) |
| 41 | \( 1 + (-0.707 + 0.707i)T \) |
| 43 | \( 1 + (-0.923 - 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.923 + 0.382i)T \) |
| 59 | \( 1 + (-0.382 + 0.923i)T \) |
| 61 | \( 1 + (0.923 - 0.382i)T \) |
| 67 | \( 1 + (-0.923 + 0.382i)T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + (-0.382 - 0.923i)T \) |
| 89 | \( 1 + (-0.707 - 0.707i)T \) |
| 97 | \( 1 + T \) |
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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
−24.89098632704483464856222100134, −23.83460570726596911681086326917, −22.971335031507894182463017092853, −22.40437515043406319623537827943, −21.47916406146385012578686261951, −20.392663165735848106227026587473, −19.296277231675683439277556252643, −18.28889795965469394658942049637, −17.92439364542898107465654308163, −16.69107933611787110688187811017, −15.7794988056277333944558001246, −15.21779033330284809733204647512, −13.389322998672437725741503933148, −12.99998546539954373250336512538, −11.954610913143270630753764447755, −11.07086588344450410404042017538, −10.076309287156288800083675651089, −9.0504652400876238479677011990, −7.62343282425388475495184972688, −6.86223004660017830959673390243, −5.53520240140147990491325936898, −5.14859997600436576703203748703, −3.33421709998110100591216473895, −2.12704980602545944467143945787, −0.344138418981639434008158697,
1.408774017011029327123254090993, 3.332731766982684298258408314009, 4.23446736249077300031069345683, 5.4613010328467274225022648296, 6.37750182431140076089900200377, 7.30160820240323199658255608615, 8.696074137695012463753242554280, 9.92977874314145081042421897984, 10.58411737312218546374066569476, 11.4484826017233090477469564605, 12.64369541053354467819142772494, 13.32470743531318833431352097533, 14.60928903679419848143662419200, 15.7170737901769335001580243505, 16.60493753523726689414429524989, 16.93287564670275112423301319577, 18.37803318712516363985054455107, 18.8796319308629023450541516704, 20.26022727281397190777989290907, 21.08327639514474995868956273896, 21.93712075911978731582046321282, 22.83661440576424708470515011246, 23.5968391723227032868271820056, 24.15723517584287515554221780305, 25.67205338369464264878437064867