L(s) = 1 | + (0.733 − 0.680i)3-s + (−0.988 − 0.149i)5-s + (−0.5 + 0.866i)7-s + (0.0747 − 0.997i)9-s + (−0.900 + 0.433i)11-s + (−0.365 − 0.930i)13-s + (−0.826 + 0.563i)15-s + (−0.988 + 0.149i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (−0.826 − 0.563i)23-s + (0.955 + 0.294i)25-s + (−0.623 − 0.781i)27-s + (−0.733 − 0.680i)29-s + (−0.955 + 0.294i)31-s + ⋯ |
L(s) = 1 | + (0.733 − 0.680i)3-s + (−0.988 − 0.149i)5-s + (−0.5 + 0.866i)7-s + (0.0747 − 0.997i)9-s + (−0.900 + 0.433i)11-s + (−0.365 − 0.930i)13-s + (−0.826 + 0.563i)15-s + (−0.988 + 0.149i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (−0.826 − 0.563i)23-s + (0.955 + 0.294i)25-s + (−0.623 − 0.781i)27-s + (−0.733 − 0.680i)29-s + (−0.955 + 0.294i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02302574892 - 0.3946842669i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02302574892 - 0.3946842669i\) |
\(L(1)\) |
\(\approx\) |
\(0.7183818783 - 0.2463761321i\) |
\(L(1)\) |
\(\approx\) |
\(0.7183818783 - 0.2463761321i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 \) |
good | 3 | \( 1 + (0.733 - 0.680i)T \) |
| 5 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.900 + 0.433i)T \) |
| 13 | \( 1 + (-0.365 - 0.930i)T \) |
| 17 | \( 1 + (-0.988 + 0.149i)T \) |
| 19 | \( 1 + (-0.0747 - 0.997i)T \) |
| 23 | \( 1 + (-0.826 - 0.563i)T \) |
| 29 | \( 1 + (-0.733 - 0.680i)T \) |
| 31 | \( 1 + (-0.955 + 0.294i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.365 + 0.930i)T \) |
| 59 | \( 1 + (0.623 + 0.781i)T \) |
| 61 | \( 1 + (0.955 + 0.294i)T \) |
| 67 | \( 1 + (0.0747 + 0.997i)T \) |
| 71 | \( 1 + (0.826 - 0.563i)T \) |
| 73 | \( 1 + (-0.365 - 0.930i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.733 + 0.680i)T \) |
| 89 | \( 1 + (0.733 - 0.680i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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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
−25.68409666807001978632078194739, −24.19428032987086480576519442668, −23.7040151206010122469165838564, −22.55109768314762152685629860545, −21.84683605967657657088204208767, −20.62958685136436091552489729465, −20.121341905829323453539777985377, −19.21752549921601742103542571288, −18.56494093695464725153817559077, −16.93147902929840453825013250152, −16.1485139543144950431303179892, −15.599374997230765331988390107105, −14.492725623690909872351107979577, −13.73032051566522229448452409892, −12.740118411877611879976273807503, −11.388847375637022076311916163739, −10.60821788115018408744881647665, −9.69905068940000581461065084207, −8.59064558344882502413721802234, −7.7027538462645586992411857751, −6.869550560955830216562585960016, −5.18775077190916361407571692400, −3.994675456518067580976815624697, −3.5268742267382709461569864021, −2.130740573291575533056100916892,
0.205223757254754594487492228, 2.225509605196901536522914152340, 2.97582743155521084330467374113, 4.23896920586618299454060440734, 5.57518618456871723678472689500, 6.8729685803689474415465287668, 7.728379610222435686822425877853, 8.54627461766455966813437395384, 9.40739492941499218016323788720, 10.75493833423304416665449443296, 11.97018787463360947503150844083, 12.72879698719907950760641682738, 13.27587488165788889234007124372, 14.80592240110534035192360279239, 15.36029777588921527495680825749, 16.03803336389888335077985133160, 17.656313293982273722624459286050, 18.34142437584350500731240343960, 19.24021820353590598904317509123, 19.95750554016691289123825544406, 20.57386094698895223312885014373, 21.918271286324780172780785800675, 22.79262576630340121735746265588, 23.793320910512906253986696573037, 24.42612831924637792493361646114