L(s) = 1 | + (−0.623 + 0.781i)2-s + (0.943 − 0.330i)3-s + (−0.222 − 0.974i)4-s + (−0.943 − 0.330i)5-s + (−0.330 + 0.943i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (0.781 − 0.623i)9-s + (0.846 − 0.532i)10-s + (0.974 + 0.222i)11-s + (−0.532 − 0.846i)12-s + (0.433 − 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (−0.111 − 0.993i)17-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)2-s + (0.943 − 0.330i)3-s + (−0.222 − 0.974i)4-s + (−0.943 − 0.330i)5-s + (−0.330 + 0.943i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (0.781 − 0.623i)9-s + (0.846 − 0.532i)10-s + (0.974 + 0.222i)11-s + (−0.532 − 0.846i)12-s + (0.433 − 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (−0.111 − 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7780049518 - 0.2405802023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7780049518 - 0.2405802023i\) |
\(L(1)\) |
\(\approx\) |
\(0.8541459569 - 0.04336243475i\) |
\(L(1)\) |
\(\approx\) |
\(0.8541459569 - 0.04336243475i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 113 | \( 1 \) |
good | 2 | \( 1 + (-0.623 + 0.781i)T \) |
| 3 | \( 1 + (0.943 - 0.330i)T \) |
| 5 | \( 1 + (-0.943 - 0.330i)T \) |
| 7 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (0.974 + 0.222i)T \) |
| 13 | \( 1 + (0.433 - 0.900i)T \) |
| 17 | \( 1 + (-0.111 - 0.993i)T \) |
| 19 | \( 1 + (-0.330 - 0.943i)T \) |
| 23 | \( 1 + (0.330 - 0.943i)T \) |
| 29 | \( 1 + (0.111 + 0.993i)T \) |
| 31 | \( 1 + (-0.433 + 0.900i)T \) |
| 37 | \( 1 + (-0.846 + 0.532i)T \) |
| 41 | \( 1 + (0.974 - 0.222i)T \) |
| 43 | \( 1 + (-0.993 + 0.111i)T \) |
| 47 | \( 1 + (-0.532 + 0.846i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 + (0.330 + 0.943i)T \) |
| 61 | \( 1 + (-0.974 - 0.222i)T \) |
| 67 | \( 1 + (0.532 + 0.846i)T \) |
| 71 | \( 1 + (0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 + (0.846 + 0.532i)T \) |
| 83 | \( 1 + (0.623 - 0.781i)T \) |
| 89 | \( 1 + (0.993 + 0.111i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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
−29.56287835902926340807291464410, −28.18329137703529488096046776813, −27.46762925229603489159070744365, −26.43770714341329400109795944283, −25.849419466512112451491965434017, −24.71965388673218274990039992866, −23.04184868218404022154314441071, −21.9663729764767940051082876242, −21.12440374492008298309971496519, −19.79357459314348871178171645625, −19.26957587146898572901748169440, −18.675562771981186274938513031456, −16.82597572204337652381012512137, −15.91705402564994817758918269779, −14.76608693292765804059205690065, −13.41525340284475544166184954114, −12.24241418433184768809766090686, −11.1745588491285471410323336765, −9.86999801686166506934307607223, −8.94375729695069719090115277708, −8.02508773199581625651904558379, −6.67681182496648137065581846425, −3.94133868834215986868181004602, −3.54288484166213287006471958003, −1.9472971630498126267019161290,
0.951959363717896116072025380581, 3.19217764528703314642115167888, 4.59914684285028672490426492776, 6.649049656493218628421076692581, 7.32663972272332399664434999544, 8.60386676248664697079844622571, 9.29263489574883384338888308371, 10.70087698316434271881347517542, 12.44624847206604062694810390629, 13.564090090114279054341148584794, 14.73565825960414567874676012058, 15.67162779336892907107956799805, 16.49197397232953545563205222486, 17.890160572209252257533016499991, 19.031193817689008650687132097680, 19.84570645926701351480675413111, 20.3261870184346286597415633096, 22.52281598994420297280388030075, 23.37495532083171493148816143975, 24.46607602978758335612811431107, 25.24597290582257620696477950843, 26.146032194666771515086132439714, 27.10176123660133082836425364092, 27.84128672854417768531864872856, 29.21423625563657506380910556782