L(s) = 1 | + (−0.0356 − 0.999i)2-s + (−0.858 + 0.512i)3-s + (−0.997 + 0.0712i)4-s + (−0.102 − 0.994i)5-s + (0.543 + 0.839i)6-s + (−0.0953 + 0.995i)7-s + (0.106 + 0.994i)8-s + (0.473 − 0.880i)9-s + (−0.990 + 0.137i)10-s + (0.819 − 0.572i)12-s + (−0.712 + 0.701i)13-s + (0.998 + 0.0598i)14-s + (0.597 + 0.801i)15-s + (0.989 − 0.142i)16-s + (−0.835 − 0.549i)17-s + (−0.896 − 0.442i)18-s + ⋯ |
L(s) = 1 | + (−0.0356 − 0.999i)2-s + (−0.858 + 0.512i)3-s + (−0.997 + 0.0712i)4-s + (−0.102 − 0.994i)5-s + (0.543 + 0.839i)6-s + (−0.0953 + 0.995i)7-s + (0.106 + 0.994i)8-s + (0.473 − 0.880i)9-s + (−0.990 + 0.137i)10-s + (0.819 − 0.572i)12-s + (−0.712 + 0.701i)13-s + (0.998 + 0.0598i)14-s + (0.597 + 0.801i)15-s + (0.989 − 0.142i)16-s + (−0.835 − 0.549i)17-s + (−0.896 − 0.442i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05051448515 - 0.3649328187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05051448515 - 0.3649328187i\) |
\(L(1)\) |
\(\approx\) |
\(0.5465598090 - 0.2264140461i\) |
\(L(1)\) |
\(\approx\) |
\(0.5465598090 - 0.2264140461i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (-0.0356 - 0.999i)T \) |
| 3 | \( 1 + (-0.858 + 0.512i)T \) |
| 5 | \( 1 + (-0.102 - 0.994i)T \) |
| 7 | \( 1 + (-0.0953 + 0.995i)T \) |
| 13 | \( 1 + (-0.712 + 0.701i)T \) |
| 17 | \( 1 + (-0.835 - 0.549i)T \) |
| 19 | \( 1 + (0.786 - 0.617i)T \) |
| 23 | \( 1 + (-0.756 + 0.654i)T \) |
| 29 | \( 1 + (0.276 - 0.961i)T \) |
| 31 | \( 1 + (0.127 + 0.991i)T \) |
| 37 | \( 1 + (0.995 - 0.0942i)T \) |
| 41 | \( 1 + (-0.978 - 0.207i)T \) |
| 43 | \( 1 + (0.895 - 0.444i)T \) |
| 47 | \( 1 + (0.247 - 0.968i)T \) |
| 53 | \( 1 + (-0.859 + 0.510i)T \) |
| 59 | \( 1 + (0.00805 + 0.999i)T \) |
| 61 | \( 1 + (-0.995 - 0.0988i)T \) |
| 67 | \( 1 + (0.905 + 0.423i)T \) |
| 71 | \( 1 + (0.0218 + 0.999i)T \) |
| 73 | \( 1 + (0.954 - 0.296i)T \) |
| 79 | \( 1 + (0.999 - 0.0414i)T \) |
| 83 | \( 1 + (-0.984 + 0.176i)T \) |
| 89 | \( 1 + (0.868 - 0.495i)T \) |
| 97 | \( 1 + (-0.930 + 0.366i)T \) |
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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
−17.95395802480716258531580590755, −17.3358589398728244169331778372, −16.784675824417753495221859254434, −16.14003908347585887548375985981, −15.48567256805925102489926636156, −14.73850757515347136328391985670, −14.089451146464615901888138516485, −13.61190183887228558671620109153, −12.77367844784198432490191896921, −12.330557010524575563133946979190, −11.215040435443770301909518743168, −10.73701437263508154746453505611, −10.02981788434532718658429634854, −9.5856370176901658806452249266, −8.1256425701569718972248850297, −7.78574864885733868711279581752, −7.23421496236844854772762207357, −6.373021286017573463629056649148, −6.27051418661411697396757213128, −5.24357404228543240177076009725, −4.51807856353315450835000338274, −3.839863611229752185530725265506, −2.90760128645031744778003722765, −1.77835292790055498268571298686, −0.72260881607469808686111344666,
0.16587127087205162396450556840, 1.126155316086522256265963973248, 2.03280882680180101124820053125, 2.76305901373054036618238673700, 3.820981573220148195000787607865, 4.455618904907676138344136165516, 5.06827841629260871236398921358, 5.48606416538359974488147944561, 6.368230216696268759270433874539, 7.398870077454469560256777153264, 8.37282978531243591268453610333, 9.13074546086655483207903328841, 9.44256005748788076626395231049, 9.99224617797008612925220746831, 11.00998759773486088497617859200, 11.64208026909249019872833850834, 12.03840891817486985611646911809, 12.40377464631101647179466695719, 13.33849219526886828602691161648, 13.85700572920125277530419572775, 14.88240994023860345397065116442, 15.71332527525541133987070945475, 16.00366463754801309985462996361, 16.99428235735363286744666880316, 17.40272037851213660117496340315