L(s) = 1 | + (0.669 + 0.743i)3-s + (−0.104 + 0.994i)9-s + (0.104 + 0.994i)11-s + (−0.809 − 0.587i)13-s + (0.978 + 0.207i)17-s + (−0.669 + 0.743i)19-s + (−0.913 − 0.406i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.669 + 0.743i)33-s + (−0.104 + 0.994i)37-s + (−0.104 − 0.994i)39-s + (−0.809 − 0.587i)41-s + 43-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)3-s + (−0.104 + 0.994i)9-s + (0.104 + 0.994i)11-s + (−0.809 − 0.587i)13-s + (0.978 + 0.207i)17-s + (−0.669 + 0.743i)19-s + (−0.913 − 0.406i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.669 + 0.743i)33-s + (−0.104 + 0.994i)37-s + (−0.104 − 0.994i)39-s + (−0.809 − 0.587i)41-s + 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3167930328 + 1.235220452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3167930328 + 1.235220452i\) |
\(L(1)\) |
\(\approx\) |
\(1.001687194 + 0.5058297582i\) |
\(L(1)\) |
\(\approx\) |
\(1.001687194 + 0.5058297582i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.669 + 0.743i)T \) |
| 11 | \( 1 + (0.104 + 0.994i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.913 - 0.406i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (-0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.978 - 0.207i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.104 + 0.994i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.913 + 0.406i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−20.32456675814888382916400888526, −19.56383066473929202345035308843, −19.07754920304233599845388914641, −18.41692403993551557028176332332, −17.50746259491337586834083113, −16.78753173500800700025978407027, −15.9268287275496453492487127448, −14.94334044671142173795104276747, −14.27737314249377642686396590075, −13.70301605396865811682630523585, −12.894344315340640915344509272722, −12.06216848196035138920605072565, −11.46198902247432580786529129111, −10.35979647374116675779283791121, −9.3487441885739666826258739317, −8.834451458186838400303247275307, −7.80675758311818206616424917487, −7.31042800446796913030814378161, −6.28598486113612180007125631865, −5.58352045404106681401397962313, −4.28143003326153927293912110229, −3.416645046898306969080542503888, −2.51170118996737373356391943257, −1.670584202213856435344679155748, −0.410696712330584271169153141457,
1.63288881431153116625966818745, 2.47201755660662670432757485908, 3.47183522820041810405025861849, 4.24449668837972931782311306142, 5.09261330054374224735617823893, 5.91955532854537313272687388401, 7.2715769402553911380030730714, 7.807705362871167858970207900670, 8.717358960880700733078231098210, 9.572860788047886504113631124851, 10.24824696424666456360613064967, 10.7279342829731631714959083411, 12.18926478559514733828406620913, 12.50405804589063497217848318093, 13.67114334315865732205646009406, 14.503955590514428495467046794769, 14.939379904275262902515411575210, 15.63622454586001482472916052229, 16.69979214017096027110114259212, 17.04037826178008692622734643046, 18.217060479540391534476228889075, 18.91723241523659313027280042312, 19.89524258908990563731047934153, 20.28971230358097688623056030223, 20.994875319453920834622688409237