L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.0581 − 0.998i)3-s + (0.766 − 0.642i)4-s + (0.993 + 0.116i)5-s + (0.396 + 0.918i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.973 + 0.230i)10-s + (−0.973 − 0.230i)11-s + (−0.686 − 0.727i)12-s + (0.396 + 0.918i)13-s + (−0.286 + 0.957i)14-s + (0.0581 − 0.998i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.0581 − 0.998i)3-s + (0.766 − 0.642i)4-s + (0.993 + 0.116i)5-s + (0.396 + 0.918i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.973 + 0.230i)10-s + (−0.973 − 0.230i)11-s + (−0.686 − 0.727i)12-s + (0.396 + 0.918i)13-s + (−0.286 + 0.957i)14-s + (0.0581 − 0.998i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.094154193 + 0.1541829005i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.094154193 + 0.1541829005i\) |
\(L(1)\) |
\(\approx\) |
\(0.7929573712 - 0.1020218959i\) |
\(L(1)\) |
\(\approx\) |
\(0.7929573712 - 0.1020218959i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.0581 - 0.998i)T \) |
| 5 | \( 1 + (0.993 + 0.116i)T \) |
| 7 | \( 1 + (0.597 - 0.802i)T \) |
| 11 | \( 1 + (-0.973 - 0.230i)T \) |
| 13 | \( 1 + (0.396 + 0.918i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.973 + 0.230i)T \) |
| 31 | \( 1 + (0.835 + 0.549i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.993 - 0.116i)T \) |
| 53 | \( 1 + (0.686 + 0.727i)T \) |
| 59 | \( 1 + (-0.0581 + 0.998i)T \) |
| 61 | \( 1 + (0.686 + 0.727i)T \) |
| 67 | \( 1 + (-0.973 - 0.230i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.686 + 0.727i)T \) |
| 79 | \( 1 + (-0.286 - 0.957i)T \) |
| 83 | \( 1 + (0.396 + 0.918i)T \) |
| 89 | \( 1 + (0.0581 + 0.998i)T \) |
| 97 | \( 1 + (-0.893 - 0.448i)T \) |
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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
−18.23841545531314443582527833789, −17.73950962057034433729391725791, −17.363157685592016390389502208598, −16.3897624888538606665272082345, −15.74647180765647484941716186144, −15.38416587859560905524047579601, −14.55392582102453488755253123455, −13.57520531729931488776677548936, −12.9347529567250952785163691217, −11.97788755215387363778641091385, −11.37115829984462462986545790959, −10.631592007072763800987741406158, −10.121528844833955758781183322433, −9.50696560052038591901969218558, −8.866281738344197624720857355356, −8.245144532802132731623801605646, −7.56878050747450289243094900639, −6.32069753262087394193708340965, −5.66579679483458016152277937165, −5.115023170829870221772207137641, −4.140102105388281780060784806874, −2.9754528848688676157181729585, −2.51807431446971703184003765161, −1.77879320287419242086270575725, −0.47857441453683779717942658972,
0.89726320516442252271403999300, 1.619653054340747641826786292725, 2.16958362415695329272207737292, 3.00898831169384876872646005910, 4.4031331347538527550989982863, 5.58817118117118694898454516718, 5.80975893959735528410885116238, 6.89578298677761329803938052651, 7.18804267841051527182423239218, 8.005941484602991233972099635632, 8.6951638550864045919499505288, 9.3250568880552966697761458218, 10.33257046730457762938011551786, 10.75689940692643750109401517841, 11.47856489999926778422656033932, 12.18654006859173106859784814511, 13.45240792987260895465532259789, 13.626445487314481915724318465482, 14.25491055604530298842859749858, 15.09689339176832859027343981578, 16.0830249495155196400089995251, 16.66123426393513695923271490714, 17.38959453198594276201115283840, 17.904992503875617821882515243257, 18.230013962670307227537026040454