L(s) = 1 | + (−0.345 − 0.938i)2-s + (−0.761 + 0.647i)4-s + (−0.595 + 0.803i)5-s + (0.625 − 0.780i)7-s + (0.870 + 0.491i)8-s + (0.959 + 0.281i)10-s + (0.851 + 0.524i)13-s + (−0.948 − 0.318i)14-s + (0.161 − 0.986i)16-s + (−0.516 + 0.856i)17-s + (−0.0855 + 0.996i)19-s + (−0.0665 − 0.997i)20-s + (0.0475 + 0.998i)23-s + (−0.290 − 0.956i)25-s + (0.198 − 0.980i)26-s + ⋯ |
L(s) = 1 | + (−0.345 − 0.938i)2-s + (−0.761 + 0.647i)4-s + (−0.595 + 0.803i)5-s + (0.625 − 0.780i)7-s + (0.870 + 0.491i)8-s + (0.959 + 0.281i)10-s + (0.851 + 0.524i)13-s + (−0.948 − 0.318i)14-s + (0.161 − 0.986i)16-s + (−0.516 + 0.856i)17-s + (−0.0855 + 0.996i)19-s + (−0.0665 − 0.997i)20-s + (0.0475 + 0.998i)23-s + (−0.290 − 0.956i)25-s + (0.198 − 0.980i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6941580229 + 0.6901644605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6941580229 + 0.6901644605i\) |
\(L(1)\) |
\(\approx\) |
\(0.7890497528 - 0.09392962083i\) |
\(L(1)\) |
\(\approx\) |
\(0.7890497528 - 0.09392962083i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.345 - 0.938i)T \) |
| 5 | \( 1 + (-0.595 + 0.803i)T \) |
| 7 | \( 1 + (0.625 - 0.780i)T \) |
| 13 | \( 1 + (0.851 + 0.524i)T \) |
| 17 | \( 1 + (-0.516 + 0.856i)T \) |
| 19 | \( 1 + (-0.0855 + 0.996i)T \) |
| 23 | \( 1 + (0.0475 + 0.998i)T \) |
| 29 | \( 1 + (0.683 + 0.730i)T \) |
| 31 | \( 1 + (-0.432 - 0.901i)T \) |
| 37 | \( 1 + (0.610 + 0.791i)T \) |
| 41 | \( 1 + (0.969 - 0.244i)T \) |
| 43 | \( 1 + (-0.580 + 0.814i)T \) |
| 47 | \( 1 + (-0.532 + 0.846i)T \) |
| 53 | \( 1 + (0.774 - 0.633i)T \) |
| 59 | \( 1 + (0.272 + 0.962i)T \) |
| 61 | \( 1 + (-0.640 - 0.768i)T \) |
| 67 | \( 1 + (0.928 + 0.371i)T \) |
| 71 | \( 1 + (-0.921 - 0.389i)T \) |
| 73 | \( 1 + (0.998 + 0.0570i)T \) |
| 79 | \( 1 + (0.217 - 0.976i)T \) |
| 83 | \( 1 + (-0.964 + 0.263i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.595 - 0.803i)T \) |
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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
−21.027021175021003484921502527819, −20.10742165016531308944836016864, −19.47889066282141959126004576105, −18.3627877486645732189761238159, −18.02929627376914832465009449494, −17.11522925054431516489553196720, −16.201689746030579051437884930963, −15.66057098170010543110731082837, −15.10821051223190278612287489156, −14.12300558162146682913894428193, −13.26518831620083543401328707775, −12.492851822189045769584287331418, −11.468558249717574118943973590530, −10.72484540401287141422953742482, −9.45517075446977521469155077399, −8.70871877719851590300836323807, −8.34447564702936177533824468167, −7.39420918012413741194797365610, −6.44436296083118590710091278058, −5.414312046881852729414610291633, −4.84138099292175737657139871169, −3.99175845146713738602234022485, −2.48895358939834876201182705484, −1.11306618917462727790764453237, −0.27913798850037533706707478712,
1.1296738806638673252575071181, 1.927212933304417276778115950356, 3.20200241721967283709947390195, 3.923437001901534759902827476227, 4.52267592157055336638139508857, 5.99712000786968095675477785830, 7.09009182596055365500366111706, 7.93745145254241253674635051050, 8.48643997655503037589496807874, 9.6883245901572653949567213891, 10.483553469107327544657048459063, 11.17882413932573197354390693868, 11.55451905477888936348128357733, 12.66253929446807636642000018180, 13.53520688361571387801580573447, 14.2524720707067050413229448337, 14.999833928261522675843311202185, 16.17109643948507937946434485160, 16.91943264809961077341853083378, 17.877721904476597659572648846141, 18.33694838095035046394419061650, 19.26333212141090475362702077696, 19.770321448372423395908987343993, 20.62532487666345046170491980986, 21.32944459565343562065213481548