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 | \( 1 + (-0.893 + 0.448i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.230013962670307227537026040454, −17.904992503875617821882515243257, −17.38959453198594276201115283840, −16.66123426393513695923271490714, −16.0830249495155196400089995251, −15.09689339176832859027343981578, −14.25491055604530298842859749858, −13.626445487314481915724318465482, −13.45240792987260895465532259789, −12.18654006859173106859784814511, −11.47856489999926778422656033932, −10.75689940692643750109401517841, −10.33257046730457762938011551786, −9.3250568880552966697761458218, −8.6951638550864045919499505288, −8.005941484602991233972099635632, −7.18804267841051527182423239218, −6.89578298677761329803938052651, −5.80975893959735528410885116238, −5.58817118117118694898454516718, −4.4031331347538527550989982863, −3.00898831169384876872646005910, −2.16958362415695329272207737292, −1.619653054340747641826786292725, −0.89726320516442252271403999300,
0.47857441453683779717942658972, 1.77879320287419242086270575725, 2.51807431446971703184003765161, 2.9754528848688676157181729585, 4.140102105388281780060784806874, 5.115023170829870221772207137641, 5.66579679483458016152277937165, 6.32069753262087394193708340965, 7.56878050747450289243094900639, 8.245144532802132731623801605646, 8.866281738344197624720857355356, 9.50696560052038591901969218558, 10.121528844833955758781183322433, 10.631592007072763800987741406158, 11.37115829984462462986545790959, 11.97788755215387363778641091385, 12.9347529567250952785163691217, 13.57520531729931488776677548936, 14.55392582102453488755253123455, 15.38416587859560905524047579601, 15.74647180765647484941716186144, 16.3897624888538606665272082345, 17.363157685592016390389502208598, 17.73950962057034433729391725791, 18.23841545531314443582527833789