L(s) = 1 | + (0.109 + 0.109i)2-s + (1.67 − 1.67i)3-s − 3.97i·4-s + (−0.861 + 4.92i)5-s + 0.367·6-s + (1.87 + 1.87i)7-s + (0.875 − 0.875i)8-s + 3.40i·9-s + (−0.635 + 0.446i)10-s − 17.1·11-s + (−6.65 − 6.65i)12-s + (1.79 − 1.79i)13-s + 0.410i·14-s + (6.79 + 9.67i)15-s − 15.7·16-s + (14.9 + 14.9i)17-s + ⋯ |
L(s) = 1 | + (0.0548 + 0.0548i)2-s + (0.557 − 0.557i)3-s − 0.993i·4-s + (−0.172 + 0.985i)5-s + 0.0612·6-s + (0.267 + 0.267i)7-s + (0.109 − 0.109i)8-s + 0.378i·9-s + (−0.0635 + 0.0446i)10-s − 1.56·11-s + (−0.554 − 0.554i)12-s + (0.137 − 0.137i)13-s + 0.0293i·14-s + (0.453 + 0.645i)15-s − 0.981·16-s + (0.880 + 0.880i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13772 - 0.233541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13772 - 0.233541i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.861 - 4.92i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 2 | \( 1 + (-0.109 - 0.109i)T + 4iT^{2} \) |
| 3 | \( 1 + (-1.67 + 1.67i)T - 9iT^{2} \) |
| 11 | \( 1 + 17.1T + 121T^{2} \) |
| 13 | \( 1 + (-1.79 + 1.79i)T - 169iT^{2} \) |
| 17 | \( 1 + (-14.9 - 14.9i)T + 289iT^{2} \) |
| 19 | \( 1 + 20.4iT - 361T^{2} \) |
| 23 | \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \) |
| 29 | \( 1 + 2.55iT - 841T^{2} \) |
| 31 | \( 1 - 14.2T + 961T^{2} \) |
| 37 | \( 1 + (35.9 + 35.9i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 23.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-7.15 + 7.15i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (6.36 + 6.36i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (10.9 - 10.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 19.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 73.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-86.3 - 86.3i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 22.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (73.4 - 73.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 55.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (52.8 - 52.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-94.8 - 94.8i)T + 9.40e3iT^{2} \) |
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
−15.82781592054645813140673024788, −14.94533413222516159623463889198, −14.02005965731327780538242995638, −12.95442036736945844821519299433, −11.01485919235026844438974888206, −10.25214781722778114703771223050, −8.317786658797831776579944514600, −7.02926929861067439315537935271, −5.34954713428449861934765447217, −2.50003234195070708078282890990,
3.37843128136223885047960577660, 4.96225976013643314528032284743, 7.66877876590620185191592992676, 8.593315251401590520642766483898, 9.943948900865475898621324855021, 11.74509745008292365907891568752, 12.80073207034368694918824036875, 13.88828493086775110691785058259, 15.50771011886671103956367169510, 16.28197842362133129018613689112