L(s) = 1 | − 0.330i·2-s − 2.69i·3-s + 1.89·4-s + (−2.12 − 0.702i)5-s − 0.890·6-s − 3.35i·7-s − 1.28i·8-s − 4.24·9-s + (−0.232 + 0.702i)10-s − 3.24·11-s − 5.08i·12-s − 1.10·14-s + (−1.89 + 5.71i)15-s + 3.35·16-s + 1.94i·17-s + 1.40i·18-s + ⋯ |
L(s) = 1 | − 0.233i·2-s − 1.55i·3-s + 0.945·4-s + (−0.949 − 0.314i)5-s − 0.363·6-s − 1.26i·7-s − 0.455i·8-s − 1.41·9-s + (−0.0734 + 0.222i)10-s − 0.978·11-s − 1.46i·12-s − 0.296·14-s + (−0.488 + 1.47i)15-s + 0.838·16-s + 0.472i·17-s + 0.331i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.203047 + 1.26019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.203047 + 1.26019i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.12 + 0.702i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.330iT - 2T^{2} \) |
| 3 | \( 1 + 2.69iT - 3T^{2} \) |
| 7 | \( 1 + 3.35iT - 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.94iT - 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 2.69iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + 1.94iT - 37T^{2} \) |
| 41 | \( 1 - 2.78T + 41T^{2} \) |
| 43 | \( 1 + 8.73iT - 43T^{2} \) |
| 47 | \( 1 - 6.86iT - 47T^{2} \) |
| 53 | \( 1 + 12.8iT - 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 - 4.01iT - 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 + 5.46iT - 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 8.61iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.26iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.980456990814754694076596870542, −8.419444080464441992744111071527, −7.66381644156780154527809898987, −7.35572005685656598806176242000, −6.62147391948760601279294069440, −5.50703811651205431388496684847, −4.02710201141742199081217455914, −2.96277233910550412854938726551, −1.68923021237348004208386702592, −0.60022131335797707425034575469,
2.60223121776646807819728451579, 3.16339827244499310259246403915, 4.47372972547060802186384204939, 5.31474117795251487297410298181, 6.13497981495027241245996216606, 7.35535032444849109014427044894, 8.190203244114939936391832372833, 8.957897036097289183093155757849, 9.965831921906862846288680197323, 10.67385043908249699917452024315