L(s) = 1 | + (0.142 + 0.989i)5-s + (−0.415 + 0.909i)7-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.841 − 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)25-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.959 − 0.281i)35-s + (−0.142 + 0.989i)37-s + (0.142 + 0.989i)41-s + (0.654 + 0.755i)43-s + 47-s + (−0.654 − 0.755i)49-s + ⋯ |
L(s) = 1 | + (0.142 + 0.989i)5-s + (−0.415 + 0.909i)7-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.841 − 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)25-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.959 − 0.281i)35-s + (−0.142 + 0.989i)37-s + (0.142 + 0.989i)41-s + (0.654 + 0.755i)43-s + 47-s + (−0.654 − 0.755i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3524363964 + 0.7356237954i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3524363964 + 0.7356237954i\) |
\(L(1)\) |
\(\approx\) |
\(0.7927825230 + 0.3557654603i\) |
\(L(1)\) |
\(\approx\) |
\(0.7927825230 + 0.3557654603i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (-0.841 - 0.540i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (-0.841 - 0.540i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 + (-0.142 + 0.989i)T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.415 + 0.909i)T \) |
| 59 | \( 1 + (0.415 + 0.909i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.959 - 0.281i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.415 - 0.909i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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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
−25.47432705983229222504492272129, −24.292559123042919763560336208083, −23.61099154196821929631287140946, −22.804665267314693545274082462621, −21.60085413083179974955365705182, −20.60111860761230821351896283764, −20.0979823044352989050933112016, −19.10345306971926571724088040877, −17.68881200647689019158587847856, −17.21709166480797923475901305112, −16.0190075430637104485499601082, −15.44042795318083164435884161493, −13.94870931363046901711896466803, −12.94684998199590809283962101123, −12.6838680506612391064961739146, −10.943986593826847569212586549589, −10.2978980981335749822413087527, −9.0563749982095197466169056830, −8.147583885865166670174111600892, −7.05622215599924839581331229268, −5.76935735836516061971207637058, −4.72175095868955078514131590421, −3.66741852951355433662506937854, −2.105870438419598806251708195572, −0.51987027990340475479586764365,
2.14179127552449307170225147886, 2.927646933367119617150026838059, 4.32712496561871511645532719014, 5.84245377939627653257439467875, 6.49123479794505518353911487699, 7.73824803034249765875044187820, 8.900803830219155376010931367367, 9.896442939000173797725827185, 10.971598407411796379924037524200, 11.75184160748279827088395989775, 13.046007032798225557851028225212, 13.85057286062243689665527012822, 15.03956869016657129809676727485, 15.644724847765060761529239910467, 16.72886057381670654134354128257, 18.04272602038707025537304506653, 18.694937234607748177629478033361, 19.25357370858250277439090198711, 20.760045213115784435673827616929, 21.52633060506236195829639655066, 22.35281118480401509751152531107, 23.15760083202472399245005078044, 24.187615108312458182193931860431, 25.235058025067596603063447641, 26.08459668949086845233493590338